topology.algebra.module.weak_dual
β·
Mathlib.Topology.Algebra.Module.WeakDual
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -253,7 +253,7 @@ instance WeakDual.instContinuousLinearMapClass : ContinuousLinearMapClass (WeakD
/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`
directly. -/
instance : CoeFun (WeakDual π E) fun _ => E β π :=
- FunLike.hasCoeToFun
+ DFunLike.hasCoeToFun
/-- If a monoid `M` distributively continuously acts on `π` and this action commutes with
multiplication on `π`, then it acts on `weak_dual π E`. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2021 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€, Moritz Doll
-/
-import Mathbin.Topology.Algebra.Module.Basic
-import Mathbin.LinearAlgebra.BilinearMap
+import Topology.Algebra.Module.Basic
+import LinearAlgebra.BilinearMap
#align_import topology.algebra.module.weak_dual from "leanprover-community/mathlib"@"25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2021 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€, Moritz Doll
-
-! This file was ported from Lean 3 source module topology.algebra.module.weak_dual
-! leanprover-community/mathlib commit 25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Topology.Algebra.Module.Basic
import Mathbin.LinearAlgebra.BilinearMap
+#align_import topology.algebra.module.weak_dual from "leanprover-community/mathlib"@"25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e"
+
/-!
# Weak dual topology
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -122,31 +122,41 @@ variable (B : E ββ[π] F ββ[π] π)
instance : TopologicalSpace (WeakBilin B) :=
TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
+#print WeakBilin.coeFn_continuous /-
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is continuous. -/
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
continuous_induced_dom
#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuous
+-/
+#print WeakBilin.eval_continuous /-
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
#align weak_bilin.eval_continuous WeakBilin.eval_continuous
+-/
+#print WeakBilin.continuous_of_continuous_eval /-
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBilin B}
(h : β y, Continuous fun a => B (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_eval
+-/
+#print WeakBilin.embedding /-
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is an embedding. -/
theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Injective B) :
Embedding fun (x : WeakBilin B) y => B x y :=
Function.Injective.embedding_induced <| LinearMap.coe_injective.comp hB
#align weak_bilin.embedding WeakBilin.embedding
+-/
+#print WeakBilin.tendsto_iff_forall_eval_tendsto /-
theorem tendsto_iff_forall_eval_tendsto {l : Filter Ξ±} {f : Ξ± β WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (π x) β β y, Tendsto (fun i => B (f i) y) l (π (B x y)) := by
rw [β tendsto_pi_nhds, Embedding.tendsto_nhds_iff (Embedding hB)]
#align weak_bilin.tendsto_iff_forall_eval_tendsto WeakBilin.tendsto_iff_forall_eval_tendsto
+-/
/-- Addition in `weak_space B` is continuous. -/
instance [ContinuousAdd π] : ContinuousAdd (WeakBilin B) :=
@@ -199,11 +209,13 @@ end WeakTopology
section WeakStarTopology
+#print topDualPairing /-
/-- The canonical pairing of a vector space and its topological dual. -/
def topDualPairing (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [AddCommMonoid E]
[Module π E] [TopologicalSpace E] [ContinuousConstSMul π π] : (E βL[π] π) ββ[π] E ββ[π] π :=
ContinuousLinearMap.coeLM π
#align top_dual_pairing topDualPairing
+-/
variable [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π]
@@ -211,11 +223,14 @@ variable [ContinuousConstSMul π π]
variable [AddCommMonoid E] [Module π E] [TopologicalSpace E]
+#print topDualPairing_apply /-
theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
rfl
#align dual_pairing_apply topDualPairing_apply
+-/
/- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] π -/
+#print WeakDual /-
/-- The weak star topology is the topology coarsest topology on `E βL[π] π` such that all
functionals `Ξ» v, top_dual_pairing π E v x` are continuous. -/
def WeakDual (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [ContinuousConstSMul π π]
@@ -225,15 +240,18 @@ deriving AddCommMonoid,
Β«./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] πΒ»,
TopologicalSpace, ContinuousAdd
#align weak_dual WeakDual
+-/
namespace WeakDual
instance : Inhabited (WeakDual π E) :=
ContinuousLinearMap.inhabited
+#print WeakDual.instContinuousLinearMapClass /-
instance WeakDual.instContinuousLinearMapClass : ContinuousLinearMapClass (WeakDual π E) π E π :=
ContinuousLinearMap.continuousSemilinearMapClass
#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.instContinuousLinearMapClass
+-/
/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`
directly. -/
@@ -252,12 +270,14 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [C
DistribMulAction M (WeakDual π E) :=
ContinuousLinearMap.distribMulAction
+#print WeakDual.instModule' /-
/-- If `π` is a topological module over a semiring `R` and scalar multiplication commutes with the
multiplication on `π`, then `weak_dual π E` is a module over `R`. -/
instance instModule' (R) [Semiring R] [Module R π] [SMulCommClass π R π] [ContinuousConstSMul R π] :
Module R (WeakDual π E) :=
ContinuousLinearMap.module
#align weak_dual.module' WeakDual.instModule'
+-/
instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [ContinuousConstSMul M π] :
ContinuousConstSMul M (WeakDual π E) :=
@@ -271,18 +291,24 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [T
β¨continuous_induced_rng.2 <|
continuous_fst.smul ((WeakBilin.coeFn_continuous (topDualPairing π E)).comp continuous_snd)β©
+#print WeakDual.coeFn_continuous /-
theorem coeFn_continuous : Continuous fun (x : WeakDual π E) y => x y :=
continuous_induced_dom
#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuous
+-/
+#print WeakDual.eval_continuous /-
theorem eval_continuous (y : E) : Continuous fun x : WeakDual π E => x y :=
continuous_pi_iff.mp coeFn_continuous y
#align weak_dual.eval_continuous WeakDual.eval_continuous
+-/
+#print WeakDual.continuous_of_continuous_eval /-
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakDual π E}
(h : β y, Continuous fun a => (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
#align weak_dual.continuous_of_continuous_eval WeakDual.continuous_of_continuous_eval
+-/
instance [T2Space π] : T2Space (WeakDual π E) :=
Embedding.t2Space <|
@@ -292,6 +318,7 @@ instance [T2Space π] : T2Space (WeakDual π E) :=
end WeakDual
/- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] π -/
+#print WeakSpace /-
/-- The weak topology is the topology coarsest topology on `E` such that all
functionals `Ξ» x, top_dual_pairing π E v x` are continuous. -/
@[nolint has_nonempty_instance]
@@ -302,11 +329,13 @@ deriving AddCommMonoid,
Β«./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] πΒ»,
TopologicalSpace, ContinuousAdd
#align weak_space WeakSpace
+-/
namespace WeakSpace
variable {π E F} [AddCommMonoid F] [Module π F] [TopologicalSpace F]
+#print WeakSpace.map /-
/-- A continuous linear map from `E` to `F` is still continuous when `E` and `F` are equipped with
their weak topologies. -/
def map (f : E βL[π] F) : WeakSpace π E βL[π] WeakSpace π F :=
@@ -314,24 +343,31 @@ def map (f : E βL[π] F) : WeakSpace π E βL[π] WeakSpace π F :=
cont :=
WeakBilin.continuous_of_continuous_eval _ fun l => WeakBilin.eval_continuous _ (l βL f) }
#align weak_space.map WeakSpace.map
+-/
+#print WeakSpace.map_apply /-
theorem map_apply (f : E βL[π] F) (x : E) : WeakSpace.map f x = f x :=
rfl
#align weak_space.map_apply WeakSpace.map_apply
+-/
+#print WeakSpace.coe_map /-
@[simp]
theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
rfl
#align weak_space.coe_map WeakSpace.coe_map
+-/
end WeakSpace
+#print tendsto_iff_forall_eval_tendsto_topDualPairing /-
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter Ξ±} {f : Ξ± β WeakDual π E}
{x : WeakDual π E} :
Tendsto f l (π x) β
β y, Tendsto (fun i => topDualPairing π E (f i) y) l (π (topDualPairing π E x y)) :=
WeakBilin.tendsto_iff_forall_eval_tendsto _ ContinuousLinearMap.coe_injective
#align tendsto_iff_forall_eval_tendsto_top_dual_pairing tendsto_iff_forall_eval_tendsto_topDualPairing
+-/
end WeakStarTopology
mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297
@@ -78,7 +78,7 @@ variable {Ξ± π π R E F M : Type _}
section WeakTopology
-/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
+/- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] π -/
#print WeakBilin /-
/-- The space `E` equipped with the weak topology induced by the bilinear form `B`. -/
@[nolint has_nonempty_instance unused_arguments]
@@ -86,7 +86,7 @@ def WeakBilin [CommSemiring π] [AddCommMonoid E] [Module π E] [AddCommMono
(B : E ββ[π] F ββ[π] π) :=
E
deriving AddCommMonoid,
- Β«./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] πΒ»
+ Β«./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] πΒ»
#align weak_bilin WeakBilin
-/
@@ -215,14 +215,14 @@ theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing
rfl
#align dual_pairing_apply topDualPairing_apply
-/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
+/- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] π -/
/-- The weak star topology is the topology coarsest topology on `E βL[π] π` such that all
functionals `Ξ» v, top_dual_pairing π E v x` are continuous. -/
def WeakDual (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [ContinuousConstSMul π π]
[AddCommMonoid E] [Module π E] [TopologicalSpace E] :=
WeakBilin (topDualPairing π E)
deriving AddCommMonoid,
- Β«./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] πΒ»,
+ Β«./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] πΒ»,
TopologicalSpace, ContinuousAdd
#align weak_dual WeakDual
@@ -291,7 +291,7 @@ instance [T2Space π] : T2Space (WeakDual π E) :=
end WeakDual
-/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
+/- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] π -/
/-- The weak topology is the topology coarsest topology on `E` such that all
functionals `Ξ» x, top_dual_pairing π E v x` are continuous. -/
@[nolint has_nonempty_instance]
@@ -299,7 +299,7 @@ def WeakSpace (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAd
[ContinuousConstSMul π π] [AddCommMonoid E] [Module π E] [TopologicalSpace E] :=
WeakBilin (topDualPairing π E).flip
deriving AddCommMonoid,
- Β«./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] πΒ»,
+ Β«./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler module[module] πΒ»,
TopologicalSpace, ContinuousAdd
#align weak_space WeakSpace
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -84,7 +84,8 @@ section WeakTopology
@[nolint has_nonempty_instance unused_arguments]
def WeakBilin [CommSemiring π] [AddCommMonoid E] [Module π E] [AddCommMonoid F] [Module π F]
(B : E ββ[π] F ββ[π] π) :=
- E deriving AddCommMonoid,
+ E
+deriving AddCommMonoid,
Β«./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] πΒ»
#align weak_bilin WeakBilin
-/
@@ -219,7 +220,8 @@ theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing
functionals `Ξ» v, top_dual_pairing π E v x` are continuous. -/
def WeakDual (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [ContinuousConstSMul π π]
[AddCommMonoid E] [Module π E] [TopologicalSpace E] :=
- WeakBilin (topDualPairing π E)deriving AddCommMonoid,
+ WeakBilin (topDualPairing π E)
+deriving AddCommMonoid,
Β«./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] πΒ»,
TopologicalSpace, ContinuousAdd
#align weak_dual WeakDual
@@ -295,7 +297,8 @@ functionals `Ξ» x, top_dual_pairing π E v x` are continuous. -/
@[nolint has_nonempty_instance]
def WeakSpace (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π]
[ContinuousConstSMul π π] [AddCommMonoid E] [Module π E] [TopologicalSpace E] :=
- WeakBilin (topDualPairing π E).flip deriving AddCommMonoid,
+ WeakBilin (topDualPairing π E).flip
+deriving AddCommMonoid,
Β«./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] πΒ»,
TopologicalSpace, ContinuousAdd
#align weak_space WeakSpace
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -72,7 +72,7 @@ noncomputable section
open Filter
-open Topology
+open scoped Topology
variable {Ξ± π π R E F M : Type _}
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -121,41 +121,26 @@ variable (B : E ββ[π] F ββ[π] π)
instance : TopologicalSpace (WeakBilin B) :=
TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
-/- warning: weak_bilin.coe_fn_continuous -> WeakBilin.coeFn_continuous is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuousβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is continuous. -/
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
continuous_induced_dom
#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuous
-/- warning: weak_bilin.eval_continuous -> WeakBilin.eval_continuous is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_bilin.eval_continuous WeakBilin.eval_continuousβ'. -/
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
#align weak_bilin.eval_continuous WeakBilin.eval_continuous
-/- warning: weak_bilin.continuous_of_continuous_eval -> WeakBilin.continuous_of_continuous_eval is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBilin B}
(h : β y, Continuous fun a => B (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_eval
-/- warning: weak_bilin.embedding -> WeakBilin.embedding is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_bilin.embedding WeakBilin.embeddingβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is an embedding. -/
theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Injective B) :
Embedding fun (x : WeakBilin B) y => B x y :=
Function.Injective.embedding_induced <| LinearMap.coe_injective.comp hB
#align weak_bilin.embedding WeakBilin.embedding
-/- warning: weak_bilin.tendsto_iff_forall_eval_tendsto -> WeakBilin.tendsto_iff_forall_eval_tendsto is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_bilin.tendsto_iff_forall_eval_tendsto WeakBilin.tendsto_iff_forall_eval_tendstoβ'. -/
theorem tendsto_iff_forall_eval_tendsto {l : Filter Ξ±} {f : Ξ± β WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (π x) β β y, Tendsto (fun i => B (f i) y) l (π (B x y)) := by
@@ -213,12 +198,6 @@ end WeakTopology
section WeakStarTopology
-/- warning: top_dual_pairing -> topDualPairing is a dubious translation:
-lean 3 declaration is
- forall (π : Type.{u1}) (E : Type.{u2}) [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : AddCommMonoid.{u2} E] [_inst_5 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))], LinearMap.{u1, u1, max u2 u1, max u2 u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (LinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E π _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u1, u1, u2, u1} π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (ContinuousLinearMap.module.{u1, u1, u1, u2, u1} π π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_6 _inst_4 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (topDualPairing._proof_1.{u1} π _inst_1) _inst_7 (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) _inst_3) (LinearMap.module.{u1, u1, u1, u2, u1} π π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (CommSemiring.toSemiring.{u1} π _inst_1) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (topDualPairing._proof_2.{u1} π _inst_1))
-but is expected to have type
- forall (π : Type.{u1}) (E : Type.{u2}) [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : AddCommMonoid.{u2} E] [_inst_5 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))], LinearMap.{u1, u1, max u1 u2, max u1 u2} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (LinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E π _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u1, u1, u2, u1} π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (ContinuousLinearMap.module.{u1, u1, u1, u2, u1} π π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_6 _inst_4 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_1) (MulActionWithZero.toMulAction.{u1, u1} π π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u1} π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))) _inst_7 (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u1, u1, u1, u2, u1} π π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (CommSemiring.toSemiring.{u1} π _inst_1) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_1) (MulActionWithZero.toMulAction.{u1, u1} π π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u1} π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))
-Case conversion may be inaccurate. Consider using '#align top_dual_pairing topDualPairingβ'. -/
/-- The canonical pairing of a vector space and its topological dual. -/
def topDualPairing (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [AddCommMonoid E]
[Module π E] [TopologicalSpace E] [ContinuousConstSMul π π] : (E βL[π] π) ββ[π] E ββ[π] π :=
@@ -231,20 +210,11 @@ variable [ContinuousConstSMul π π]
variable [AddCommMonoid E] [Module π E] [TopologicalSpace E]
-/- warning: dual_pairing_apply -> topDualPairing_apply is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align dual_pairing_apply topDualPairing_applyβ'. -/
theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
rfl
#align dual_pairing_apply topDualPairing_apply
/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
-/- warning: weak_dual -> WeakDual is a dubious translation:
-lean 3 declaration is
- forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8))))))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{max u2 u1}
-but is expected to have type
- forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Algebra.toSMul.{u1, u1} π π _inst_8 (CommSemiring.toSemiring.{u1} π _inst_8) (Algebra.id.{u1} π _inst_8))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{max u1 u2}
-Case conversion may be inaccurate. Consider using '#align weak_dual WeakDualβ'. -/
/-- The weak star topology is the topology coarsest topology on `E βL[π] π` such that all
functionals `Ξ» v, top_dual_pairing π E v x` are continuous. -/
def WeakDual (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [ContinuousConstSMul π π]
@@ -259,12 +229,6 @@ namespace WeakDual
instance : Inhabited (WeakDual π E) :=
ContinuousLinearMap.inhabited
-/- warning: weak_dual.weak_dual.continuous_linear_map_class -> WeakDual.instContinuousLinearMapClass is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E], ContinuousLinearMapClass.{max u2 u1, u1, u2, u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))
-but is expected to have type
- forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E], ContinuousLinearMapClass.{max u2 u1, u1, u2, u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))
-Case conversion may be inaccurate. Consider using '#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.instContinuousLinearMapClassβ'. -/
instance WeakDual.instContinuousLinearMapClass : ContinuousLinearMapClass (WeakDual π E) π E π :=
ContinuousLinearMap.continuousSemilinearMapClass
#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.instContinuousLinearMapClass
@@ -286,12 +250,6 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [C
DistribMulAction M (WeakDual π E) :=
ContinuousLinearMap.distribMulAction
-/- warning: weak_dual.module' -> WeakDual.instModule' is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (R : Type.{u3}) [_inst_8 : Semiring.{u3} R] [_inst_9 : Module.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))] [_inst_10 : SMulCommClass.{u1, u3, u1} π R π (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))) (SMulZeroClass.toHasSmul.{u3, u1} R π (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (SMulWithZero.toSmulZeroClass.{u3, u1} R π (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)))) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))] [_inst_11 : ContinuousConstSMul.{u3, u1} R π _inst_2 (SMulZeroClass.toHasSmul.{u3, u1} R π (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (SMulWithZero.toSmulZeroClass.{u3, u1} R π (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)))) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))], Module.{u3, max u2 u1} R (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)
-but is expected to have type
- forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (R : Type.{u3}) [_inst_8 : Semiring.{u3} R] [_inst_9 : Module.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))] [_inst_10 : SMulCommClass.{u1, u3, u1} π R π (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1)) (SMulZeroClass.toSMul.{u3, u1} R π (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (SMulWithZero.toSMulZeroClass.{u3, u1} R π (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))] [_inst_11 : ContinuousConstSMul.{u3, u1} R π _inst_2 (SMulZeroClass.toSMul.{u3, u1} R π (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (SMulWithZero.toSMulZeroClass.{u3, u1} R π (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))], Module.{u3, max u2 u1} R (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.instAddCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)
-Case conversion may be inaccurate. Consider using '#align weak_dual.module' WeakDual.instModule'β'. -/
/-- If `π` is a topological module over a semiring `R` and scalar multiplication commutes with the
multiplication on `π`, then `weak_dual π E` is a module over `R`. -/
instance instModule' (R) [Semiring R] [Module R π] [SMulCommClass π R π] [ContinuousConstSMul R π] :
@@ -311,23 +269,14 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [T
β¨continuous_induced_rng.2 <|
continuous_fst.smul ((WeakBilin.coeFn_continuous (topDualPairing π E)).comp continuous_snd)β©
-/- warning: weak_dual.coe_fn_continuous -> WeakDual.coeFn_continuous is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuousβ'. -/
theorem coeFn_continuous : Continuous fun (x : WeakDual π E) y => x y :=
continuous_induced_dom
#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuous
-/- warning: weak_dual.eval_continuous -> WeakDual.eval_continuous is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_dual.eval_continuous WeakDual.eval_continuousβ'. -/
theorem eval_continuous (y : E) : Continuous fun x : WeakDual π E => x y :=
continuous_pi_iff.mp coeFn_continuous y
#align weak_dual.eval_continuous WeakDual.eval_continuous
-/- warning: weak_dual.continuous_of_continuous_eval -> WeakDual.continuous_of_continuous_eval is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_dual.continuous_of_continuous_eval WeakDual.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakDual π E}
(h : β y, Continuous fun a => (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
@@ -341,12 +290,6 @@ instance [T2Space π] : T2Space (WeakDual π E) :=
end WeakDual
/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
-/- warning: weak_space -> WeakSpace is a dubious translation:
-lean 3 declaration is
- forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8))))))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{u2}
-but is expected to have type
- forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Algebra.toSMul.{u1, u1} π π _inst_8 (CommSemiring.toSemiring.{u1} π _inst_8) (Algebra.id.{u1} π _inst_8))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{u2}
-Case conversion may be inaccurate. Consider using '#align weak_space WeakSpaceβ'. -/
/-- The weak topology is the topology coarsest topology on `E` such that all
functionals `Ξ» x, top_dual_pairing π E v x` are continuous. -/
@[nolint has_nonempty_instance]
@@ -361,9 +304,6 @@ namespace WeakSpace
variable {π E F} [AddCommMonoid F] [Module π F] [TopologicalSpace F]
-/- warning: weak_space.map -> WeakSpace.map is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_space.map WeakSpace.mapβ'. -/
/-- A continuous linear map from `E` to `F` is still continuous when `E` and `F` are equipped with
their weak topologies. -/
def map (f : E βL[π] F) : WeakSpace π E βL[π] WeakSpace π F :=
@@ -372,16 +312,10 @@ def map (f : E βL[π] F) : WeakSpace π E βL[π] WeakSpace π F :=
WeakBilin.continuous_of_continuous_eval _ fun l => WeakBilin.eval_continuous _ (l βL f) }
#align weak_space.map WeakSpace.map
-/- warning: weak_space.map_apply -> WeakSpace.map_apply is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_space.map_apply WeakSpace.map_applyβ'. -/
theorem map_apply (f : E βL[π] F) (x : E) : WeakSpace.map f x = f x :=
rfl
#align weak_space.map_apply WeakSpace.map_apply
-/- warning: weak_space.coe_map -> WeakSpace.coe_map is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align weak_space.coe_map WeakSpace.coe_mapβ'. -/
@[simp]
theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
rfl
@@ -389,9 +323,6 @@ theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
end WeakSpace
-/- warning: tendsto_iff_forall_eval_tendsto_top_dual_pairing -> tendsto_iff_forall_eval_tendsto_topDualPairing is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align tendsto_iff_forall_eval_tendsto_top_dual_pairing tendsto_iff_forall_eval_tendsto_topDualPairingβ'. -/
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter Ξ±} {f : Ξ± β WeakDual π E}
{x : WeakDual π E} :
Tendsto f l (π x) β
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -122,10 +122,7 @@ instance : TopologicalSpace (WeakBilin B) :=
TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
/- warning: weak_bilin.coe_fn_continuous -> WeakBilin.coeFn_continuous is a dubious translation:
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(Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π 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+<too large>
Case conversion may be inaccurate. Consider using '#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuousβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is continuous. -/
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
@@ -133,20 +130,14 @@ theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuous
/- warning: weak_bilin.eval_continuous -> WeakBilin.eval_continuous is a dubious translation:
-lean 3 declaration is
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(Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) (y : F), Continuous.{u2, u1} (WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) π (WeakBilin.topologicalSpace.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_1 (fun (x : WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) => coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π 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Case conversion may be inaccurate. Consider using '#align weak_bilin.eval_continuous WeakBilin.eval_continuousβ'. -/
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
#align weak_bilin.eval_continuous WeakBilin.eval_continuous
/- warning: weak_bilin.continuous_of_continuous_eval -> WeakBilin.continuous_of_continuous_eval is a dubious translation:
-lean 3 declaration is
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+<too large>
Case conversion may be inaccurate. Consider using '#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBilin B}
(h : β y, Continuous fun a => B (g a) y) : Continuous g :=
@@ -154,10 +145,7 @@ theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBili
#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_eval
/- warning: weak_bilin.embedding -> WeakBilin.embedding is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align weak_bilin.embedding WeakBilin.embeddingβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is an embedding. -/
theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Injective B) :
@@ -166,10 +154,7 @@ theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Inject
#align weak_bilin.embedding WeakBilin.embedding
/- warning: weak_bilin.tendsto_iff_forall_eval_tendsto -> WeakBilin.tendsto_iff_forall_eval_tendsto is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align weak_bilin.tendsto_iff_forall_eval_tendsto WeakBilin.tendsto_iff_forall_eval_tendstoβ'. -/
theorem tendsto_iff_forall_eval_tendsto {l : Filter Ξ±} {f : Ξ± β WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
@@ -247,10 +232,7 @@ variable [ContinuousConstSMul π π]
variable [AddCommMonoid E] [Module π E] [TopologicalSpace E]
/- warning: dual_pairing_apply -> topDualPairing_apply is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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(CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π 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(CommSemiring.toSemiring.{u2} π _inst_1))) (fun (_x : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) v) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) v x)
+<too large>
Case conversion may be inaccurate. Consider using '#align dual_pairing_apply topDualPairing_applyβ'. -/
theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
rfl
@@ -330,30 +312,21 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [T
continuous_fst.smul ((WeakBilin.coeFn_continuous (topDualPairing π E)).comp continuous_snd)β©
/- warning: weak_dual.coe_fn_continuous -> WeakDual.coeFn_continuous is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E], Continuous.{max u2 u1, max u2 u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (E -> π) (WeakDual.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Pi.topologicalSpace.{u2, u1} E (fun (y : E) => π) (fun (a : E) => _inst_2)) (fun (x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (y : E) => coeFn.{succ (max u2 u1), max (succ u2) (succ u1)} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) x y)
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+<too large>
Case conversion may be inaccurate. Consider using '#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuousβ'. -/
theorem coeFn_continuous : Continuous fun (x : WeakDual π E) y => x y :=
continuous_induced_dom
#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuous
/- warning: weak_dual.eval_continuous -> WeakDual.eval_continuous is a dubious translation:
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- forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (y : E), Continuous.{max u2 u1, u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π (WeakDual.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_2 (fun (x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => coeFn.{succ (max u2 u1), max (succ u2) (succ u1)} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) x y)
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+<too large>
Case conversion may be inaccurate. Consider using '#align weak_dual.eval_continuous WeakDual.eval_continuousβ'. -/
theorem eval_continuous (y : E) : Continuous fun x : WeakDual π E => x y :=
continuous_pi_iff.mp coeFn_continuous y
#align weak_dual.eval_continuous WeakDual.eval_continuous
/- warning: weak_dual.continuous_of_continuous_eval -> WeakDual.continuous_of_continuous_eval is a dubious translation:
-lean 3 declaration is
- forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toHasAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Mul.toSMul.{u2} π (Distrib.toHasMul.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u3} E] [_inst_6 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u3} E] [_inst_8 : TopologicalSpace.{u1} Ξ±] {g : Ξ± -> (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)}, (forall (y : E), Continuous.{u1, u2} Ξ± π _inst_8 _inst_2 (fun (a : Ξ±) => coeFn.{succ (max u3 u2), max (succ u3) (succ u2)} (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (g a) y)) -> (Continuous.{u1, max u3 u2} Ξ± (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.topologicalSpace.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) g)
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- forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] [_inst_8 : TopologicalSpace.{u3} Ξ±] {g : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)}, (forall (y : E), Continuous.{u3, u2} Ξ± ((fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) y) _inst_8 _inst_2 (fun (a : Ξ±) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) (g a) y)) -> (Continuous.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) g)
+<too large>
Case conversion may be inaccurate. Consider using '#align weak_dual.continuous_of_continuous_eval WeakDual.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakDual π E}
(h : β y, Continuous fun a => (g a) y) : Continuous g :=
@@ -389,10 +362,7 @@ namespace WeakSpace
variable {π E F} [AddCommMonoid F] [Module π F] [TopologicalSpace F]
/- warning: weak_space.map -> WeakSpace.map is a dubious translation:
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- forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u3} F] [_inst_9 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u3} F], (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) -> (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10))
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- forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u3} F] [_inst_9 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u3} F], (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) -> (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10))
+<too large>
Case conversion may be inaccurate. Consider using '#align weak_space.map WeakSpace.mapβ'. -/
/-- A continuous linear map from `E` to `F` is still continuous when `E` and `F` are equipped with
their weak topologies. -/
@@ -403,20 +373,14 @@ def map (f : E βL[π] F) : WeakSpace π E βL[π] WeakSpace π F :=
#align weak_space.map WeakSpace.map
/- warning: weak_space.map_apply -> WeakSpace.map_apply is a dubious translation:
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_inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9))) f x)
+<too large>
Case conversion may be inaccurate. Consider using '#align weak_space.map_apply WeakSpace.map_applyβ'. -/
theorem map_apply (f : E βL[π] F) (x : E) : WeakSpace.map f x = f x :=
rfl
#align weak_space.map_apply WeakSpace.map_apply
/- warning: weak_space.coe_map -> WeakSpace.coe_map is a dubious translation:
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+<too large>
Case conversion may be inaccurate. Consider using '#align weak_space.coe_map WeakSpace.coe_mapβ'. -/
@[simp]
theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
@@ -426,10 +390,7 @@ theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
end WeakSpace
/- warning: tendsto_iff_forall_eval_tendsto_top_dual_pairing -> tendsto_iff_forall_eval_tendsto_topDualPairing is a dubious translation:
-lean 3 declaration is
- forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toHasAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Mul.toSMul.{u2} π (Distrib.toHasMul.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u3} E] [_inst_6 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u3} E] {l : Filter.{u1} Ξ±} {f : Ξ± -> (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff 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_inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
+<too large>
Case conversion may be inaccurate. Consider using '#align tendsto_iff_forall_eval_tendsto_top_dual_pairing tendsto_iff_forall_eval_tendsto_topDualPairingβ'. -/
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter Ξ±} {f : Ξ± β WeakDual π E}
{x : WeakDual π E} :
mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b
@@ -125,7 +125,7 @@ instance : TopologicalSpace (WeakBilin B) :=
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] (B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 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(Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))), Continuous.{u2, max u3 u1} (WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (F -> π) (WeakBilin.topologicalSpace.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u3, u1} F (fun (y : F) => π) (fun (a : F) => _inst_1)) (fun (x : WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (y : F) => coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} 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(RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) => E -> (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u1} π π E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) B x) y)
but is expected to have type
- forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))), Continuous.{u3, max u2 u1} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (forall (y : F), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u1, u2} F (fun (y : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) (fun (a : F) => _inst_1)) (fun (x : WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (y : F) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) x) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), succ u3, max (succ u2) (succ u1)} (LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, max u2 u1} π π E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
+ forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))), Continuous.{u3, max u2 u1} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (forall (y : F), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u1, u2} F (fun (y : F) => (fun 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(CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
Case conversion may be inaccurate. Consider using '#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuousβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is continuous. -/
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
@@ -136,7 +136,7 @@ theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] (B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 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π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) => E -> (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u1} π π E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) B x) y)
but is expected to have type
- forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) (y : F), Continuous.{u3, u2} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_1 (fun (x : WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) x) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), succ u3, max (succ u2) (succ u1)} (LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, max u2 u1} π π E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
+ forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) (y : F), Continuous.{u3, u2} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_1 (fun (x : WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) => 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Case conversion may be inaccurate. Consider using '#align weak_bilin.eval_continuous WeakBilin.eval_continuousβ'. -/
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
@@ -146,7 +146,7 @@ theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
lean 3 declaration is
forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u4} F] [_inst_6 : Module.{u2, u4} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u4 u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) 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(RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2)))))) [_inst_7 : TopologicalSpace.{u1} Ξ±] {g : Ξ± -> (WeakBilin.{u2, u3, u4} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)}, (forall (y : F), Continuous.{u1, u2} Ξ± π _inst_7 _inst_1 (fun (a : Ξ±) => coeFn.{max (succ u4) (succ u2), max (succ u4) (succ u2)} (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π 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u2, u4, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.module.{u2, u2, u2, u4, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2)))))) => E -> (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (LinearMap.hasCoeToFun.{u2, u2, u3, max u4 u2} π π E (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u4, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.module.{u2, u2, u2, u4, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B (g a)) y)) -> (Continuous.{u1, u3} Ξ± (WeakBilin.{u2, u3, u4} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_7 (WeakBilin.topologicalSpace.{u2, u3, u4} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) g)
but is expected to have type
- forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) [_inst_7 : TopologicalSpace.{u4} Ξ±] {g : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)}, (forall (y : F), Continuous.{u4, u3} Ξ± ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) _inst_7 _inst_1 (fun (a : Ξ±) => FunLike.coe.{max (succ u3) (succ u1), succ u1, succ u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (g a)) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u2, max (succ u3) (succ u1)} (LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B (g a)) y)) -> (Continuous.{u4, u2} Ξ± (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_7 (WeakBilin.instTopologicalSpace.{u3, u2, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) g)
+ forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) [_inst_7 : TopologicalSpace.{u4} Ξ±] {g : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)}, (forall (y : F), Continuous.{u4, u3} Ξ± ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : F) => π) y) _inst_7 _inst_1 (fun (a : Ξ±) => FunLike.coe.{max (succ u3) (succ u1), succ u1, succ u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (g a)) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π 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Case conversion may be inaccurate. Consider using '#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBilin B}
(h : β y, Continuous fun a => B (g a) y) : Continuous g :=
@@ -157,7 +157,7 @@ theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBili
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] {B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 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but is expected to have type
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(Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π 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+ forall {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] {B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))}, (Function.Injective.{succ u2, max (succ u3) (succ u1)} E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π 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Case conversion may be inaccurate. Consider using '#align weak_bilin.embedding WeakBilin.embeddingβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is an embedding. -/
theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Injective B) :
@@ -169,7 +169,7 @@ theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Inject
lean 3 declaration is
forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u4} F] [_inst_6 : Module.{u2, u4} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u4 u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y))))
but is expected to have type
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(CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) {l : Filter.{u4} Ξ±} {f : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)} {x : WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B}, (Function.Injective.{succ u2, max (succ u3) (succ u1)} E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u2, max (succ u3) (succ u1)} (LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π 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(Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B (f i)) y) l (nhds.{u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) _inst_1 (FunLike.coe.{max (succ u3) (succ u1), succ u1, succ u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π 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_inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B x) y))))
+ forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π 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Case conversion may be inaccurate. Consider using '#align weak_bilin.tendsto_iff_forall_eval_tendsto WeakBilin.tendsto_iff_forall_eval_tendstoβ'. -/
theorem tendsto_iff_forall_eval_tendsto {l : Filter Ξ±} {f : Ξ± β WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
@@ -250,7 +250,7 @@ variable [AddCommMonoid E] [Module π E] [TopologicalSpace E]
lean 3 declaration is
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(ContinuousLinearMap.toFun.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) v x)
but is expected to have type
- forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (v : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π 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(CommSemiring.toSemiring.{u2} π _inst_1))) (fun (_x : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) v) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) v x)
+ forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (v : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (x : E), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : E) => π) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) v) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : E) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π 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(CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) v) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π 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(CommSemiring.toSemiring.{u2} π _inst_1))))) v x)
Case conversion may be inaccurate. Consider using '#align dual_pairing_apply topDualPairing_applyβ'. -/
theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
rfl
@@ -429,7 +429,7 @@ end WeakSpace
lean 3 declaration is
forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toHasAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Mul.toSMul.{u2} π (Distrib.toHasMul.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u3} E] [_inst_6 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u3} E] {l : Filter.{u1} Ξ±} {f : Ξ± -> (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff 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_inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u3, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u3, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (topDualPairing._proof_1.{u2} π _inst_1) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.module.{u2, u2, u2, u3, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (topDualPairing._proof_2.{u2} π _inst_1))) => (ContinuousLinearMap.{u2, u2, u3, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) -> (LinearMap.{u2, u2, u3, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (LinearMap.hasCoeToFun.{u2, u2, max u3 u2, max u3 u2} π π (ContinuousLinearMap.{u2, u2, u3, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u3, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u3, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u3, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u3, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (topDualPairing._proof_1.{u2} π _inst_1) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.module.{u2, u2, u2, u3, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (topDualPairing._proof_2.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
but is expected to have type
- forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] {l : Filter.{u3} Ξ±} {f : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff (Filter.Tendsto.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) f l 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(CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (f i)) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (FunLike.coe.{max (succ u2) (succ u1), max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u2, u2, max u2 u1, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) 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(CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
+ forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] {l : Filter.{u3} Ξ±} {f : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff (Filter.Tendsto.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) f l 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u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
Case conversion may be inaccurate. Consider using '#align tendsto_iff_forall_eval_tendsto_top_dual_pairing tendsto_iff_forall_eval_tendsto_topDualPairingβ'. -/
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter Ξ±} {f : Ξ± β WeakDual π E}
{x : WeakDual π E} :
mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e
@@ -125,7 +125,7 @@ instance : TopologicalSpace (WeakBilin B) :=
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] (B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))), Continuous.{u2, max u3 u1} (WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (F -> π) (WeakBilin.topologicalSpace.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u3, u1} F (fun (y : F) => π) (fun (a : F) => _inst_1)) (fun (x : WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (y : F) => coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) (coeFn.{max (succ u2) (succ (max u3 u1)), max (succ u2) (succ (max u3 u1))} (LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) => E -> (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u1} π π E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) B x) y)
but is expected to have type
- forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))), Continuous.{u3, max u2 u1} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (forall (y : F), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u1, u2} F (fun (y : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) y) (fun (a : F) => _inst_1)) (fun (x : WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (y : F) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) x) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), succ u3, max (succ u2) (succ u1)} (LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, max u2 u1} π π E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
+ forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))), Continuous.{u3, max u2 u1} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (forall (y : F), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u1, u2} F (fun (y : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) (fun (a : F) => _inst_1)) (fun (x : WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (y : F) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) x) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), succ u3, max (succ u2) (succ u1)} (LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, max u2 u1} π π E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
Case conversion may be inaccurate. Consider using '#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuousβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is continuous. -/
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
@@ -136,7 +136,7 @@ theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] (B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) (y : F), Continuous.{u2, u1} (WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) π (WeakBilin.topologicalSpace.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_1 (fun (x : WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) => coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π 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but is expected to have type
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(CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), succ u3, max (succ u2) (succ u1)} (LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} 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(Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, max u2 u1} π π E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
+ forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) (y : F), Continuous.{u3, u2} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_1 (fun (x : WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) x) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), succ u3, max (succ u2) (succ u1)} (LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, max u2 u1} π π E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
Case conversion may be inaccurate. Consider using '#align weak_bilin.eval_continuous WeakBilin.eval_continuousβ'. -/
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
@@ -146,7 +146,7 @@ theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
lean 3 declaration is
forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u4} F] [_inst_6 : Module.{u2, u4} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u4 u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u4, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.module.{u2, u2, u2, u4, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2)))))) [_inst_7 : TopologicalSpace.{u1} Ξ±] {g : Ξ± -> (WeakBilin.{u2, u3, u4} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)}, (forall (y : F), Continuous.{u1, u2} Ξ± π _inst_7 _inst_1 (fun (a : Ξ±) => coeFn.{max (succ u4) (succ u2), max (succ u4) (succ u2)} (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π 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π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u4, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.module.{u2, u2, u2, u4, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B (g a)) y)) -> (Continuous.{u1, u3} Ξ± (WeakBilin.{u2, u3, u4} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_7 (WeakBilin.topologicalSpace.{u2, u3, u4} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) g)
but is expected to have type
- forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) [_inst_7 : TopologicalSpace.{u4} Ξ±] {g : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)}, (forall (y : F), Continuous.{u4, u3} Ξ± ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) y) _inst_7 _inst_1 (fun (a : Ξ±) => FunLike.coe.{max (succ u3) (succ u1), succ u1, succ u3} ((fun 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(CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u2, max (succ u3) (succ u1)} (LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B (g a)) y)) -> (Continuous.{u4, u2} Ξ± (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_7 (WeakBilin.instTopologicalSpace.{u3, u2, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) g)
+ forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) [_inst_7 : TopologicalSpace.{u4} Ξ±] {g : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)}, (forall (y : F), Continuous.{u4, u3} Ξ± ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) y) _inst_7 _inst_1 (fun (a : Ξ±) => FunLike.coe.{max (succ u3) (succ u1), succ u1, succ u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (g a)) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u2, max (succ u3) (succ u1)} (LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B (g a)) y)) -> (Continuous.{u4, u2} Ξ± (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_7 (WeakBilin.instTopologicalSpace.{u3, u2, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) g)
Case conversion may be inaccurate. Consider using '#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBilin B}
(h : β y, Continuous fun a => B (g a) y) : Continuous g :=
@@ -157,7 +157,7 @@ theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBili
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] {B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) B x) y))
but is expected to have type
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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B x) y))
+ forall {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] {B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))}, (Function.Injective.{succ u2, max (succ u3) (succ u1)} E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B x) y))
Case conversion may be inaccurate. Consider using '#align weak_bilin.embedding WeakBilin.embeddingβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is an embedding. -/
theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Injective B) :
@@ -169,7 +169,7 @@ theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Inject
lean 3 declaration is
forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u4} F] [_inst_6 : Module.{u2, u4} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u4 u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y))))
but is expected to have type
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(CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π 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+ forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) {l : Filter.{u4} Ξ±} {f : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)} {x : WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B}, (Function.Injective.{succ u2, max (succ u3) (succ u1)} E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u2, max (succ u3) (succ u1)} (LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π 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(Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B)) -> (Iff (Filter.Tendsto.{u4, u2} Ξ± (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) f l (nhds.{u2} (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (WeakBilin.instTopologicalSpace.{u3, u2, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) x)) (forall (y : F), Filter.Tendsto.{u4, 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Case conversion may be inaccurate. Consider using '#align weak_bilin.tendsto_iff_forall_eval_tendsto WeakBilin.tendsto_iff_forall_eval_tendstoβ'. -/
theorem tendsto_iff_forall_eval_tendsto {l : Filter Ξ±} {f : Ξ± β WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
@@ -250,7 +250,7 @@ variable [AddCommMonoid E] [Module π E] [TopologicalSpace E]
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (v : ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π 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(Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (LinearMap.hasCoeToFun.{u1, u1, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (LinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u1, u1, u2, u1} π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (ContinuousLinearMap.module.{u1, u1, u1, u2, u1} π π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (topDualPairing._proof_1.{u1} π _inst_1) _inst_4 (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π 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(CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) => E -> π) (ContinuousLinearMap.toFun.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) v x)
but is expected to have type
- forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (v : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (x : E), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => π) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) v) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (FunLike.coe.{max (succ u2) (succ u1), max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u2, u2, max u2 u1, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (fun (_x : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) v) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) v x)
+ forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (v : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (x : E), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => π) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) v) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (FunLike.coe.{max (succ u2) (succ u1), max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u2, u2, max u2 u1, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π 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(CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π 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_inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (fun (_x : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) v) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) v x)
Case conversion may be inaccurate. Consider using '#align dual_pairing_apply topDualPairing_applyβ'. -/
theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
rfl
@@ -429,7 +429,7 @@ end WeakSpace
lean 3 declaration is
forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toHasAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Mul.toSMul.{u2} π (Distrib.toHasMul.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u3} E] [_inst_6 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u3} E] {l : Filter.{u1} Ξ±} {f : Ξ± -> (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (topDualPairing._proof_2.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
but is expected to have type
- forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] {l : Filter.{u3} Ξ±} {f : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff (Filter.Tendsto.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) f l 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_inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
+ forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] {l : Filter.{u3} Ξ±} {f : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff (Filter.Tendsto.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) f l 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(CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (f i)) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : E) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π 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_inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
Case conversion may be inaccurate. Consider using '#align tendsto_iff_forall_eval_tendsto_top_dual_pairing tendsto_iff_forall_eval_tendsto_topDualPairingβ'. -/
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter Ξ±} {f : Ξ± β WeakDual π E}
{x : WeakDual π E} :
mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
@@ -333,7 +333,7 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [T
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E], Continuous.{max u2 u1, max u2 u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (E -> π) (WeakDual.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Pi.topologicalSpace.{u2, u1} E (fun (y : E) => π) (fun (a : E) => _inst_2)) (fun (x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (y : E) => coeFn.{succ (max u2 u1), max (succ u2) (succ u1)} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) x y)
but is expected to have type
- forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E], Continuous.{max u2 u1, max u2 u1} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (forall (y : E), (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Pi.topologicalSpace.{u1, u2} E (fun (y : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) (fun (a : E) => _inst_2)) (fun (x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (y : E) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) x y)
+ forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E], Continuous.{max u2 u1, max u2 u1} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (forall (y : E), (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) y) (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Pi.topologicalSpace.{u1, u2} E (fun (y : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) y) (fun (a : E) => _inst_2)) (fun (x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (y : E) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) x y)
Case conversion may be inaccurate. Consider using '#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuousβ'. -/
theorem coeFn_continuous : Continuous fun (x : WeakDual π E) y => x y :=
continuous_induced_dom
@@ -343,7 +343,7 @@ theorem coeFn_continuous : Continuous fun (x : WeakDual π E) y => x y :=
lean 3 declaration is
forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (y : E), Continuous.{max u2 u1, u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π (WeakDual.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_2 (fun (x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => coeFn.{succ (max u2 u1), max (succ u2) (succ u1)} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) x y)
but is expected to have type
- forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (y : E), Continuous.{max u2 u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) ((fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_2 (fun (x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) x y)
+ forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (y : E), Continuous.{max u2 u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) ((fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) y) (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_2 (fun (x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) x y)
Case conversion may be inaccurate. Consider using '#align weak_dual.eval_continuous WeakDual.eval_continuousβ'. -/
theorem eval_continuous (y : E) : Continuous fun x : WeakDual π E => x y :=
continuous_pi_iff.mp coeFn_continuous y
@@ -353,7 +353,7 @@ theorem eval_continuous (y : E) : Continuous fun x : WeakDual π E => x y :=
lean 3 declaration is
forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toHasAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Mul.toSMul.{u2} π (Distrib.toHasMul.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u3} E] [_inst_6 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u3} E] [_inst_8 : TopologicalSpace.{u1} Ξ±] {g : Ξ± -> (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)}, (forall (y : E), Continuous.{u1, u2} Ξ± π _inst_8 _inst_2 (fun (a : Ξ±) => coeFn.{succ (max u3 u2), max (succ u3) (succ u2)} (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (g a) y)) -> (Continuous.{u1, max u3 u2} Ξ± (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.topologicalSpace.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) g)
but is expected to have type
- forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] [_inst_8 : TopologicalSpace.{u3} Ξ±] {g : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)}, (forall (y : E), Continuous.{u3, u2} Ξ± ((fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) _inst_8 _inst_2 (fun (a : Ξ±) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) (g a) y)) -> (Continuous.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) g)
+ forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] [_inst_8 : TopologicalSpace.{u3} Ξ±] {g : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)}, (forall (y : E), Continuous.{u3, u2} Ξ± ((fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) y) _inst_8 _inst_2 (fun (a : Ξ±) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1927 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) (g a) y)) -> (Continuous.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) g)
Case conversion may be inaccurate. Consider using '#align weak_dual.continuous_of_continuous_eval WeakDual.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakDual π E}
(h : β y, Continuous fun a => (g a) y) : Continuous g :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€, Moritz Doll
! This file was ported from Lean 3 source module topology.algebra.module.weak_dual
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.LinearAlgebra.BilinearMap
/-!
# Weak dual topology
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
This file defines the weak topology given two vector spaces `E` and `F` over a commutative semiring
`π` and a bilinear form `B : E ββ[π] F ββ[π] π`. The weak topology on `E` is the coarsest topology
such that for all `y : F` every map `Ξ» x, B x y` is continuous.
mathlib commit https://github.com/leanprover-community/mathlib/commit/e05ead7993520a432bec94ac504842d90707ad63
@@ -76,6 +76,7 @@ variable {Ξ± π π R E F M : Type _}
section WeakTopology
/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
+#print WeakBilin /-
/-- The space `E` equipped with the weak topology induced by the bilinear form `B`. -/
@[nolint has_nonempty_instance unused_arguments]
def WeakBilin [CommSemiring π] [AddCommMonoid E] [Module π E] [AddCommMonoid F] [Module π F]
@@ -83,6 +84,7 @@ def WeakBilin [CommSemiring π] [AddCommMonoid E] [Module π E] [AddCommMono
E deriving AddCommMonoid,
Β«./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] πΒ»
#align weak_bilin WeakBilin
+-/
namespace WeakBilin
@@ -90,11 +92,13 @@ instance [CommSemiring π] [a : AddCommGroup E] [Module π E] [AddCommMonoid
(B : E ββ[π] F ββ[π] π) : AddCommGroup (WeakBilin B) :=
a
-instance (priority := 100) module' [CommSemiring π] [CommSemiring π] [AddCommGroup E] [Module π E]
- [AddCommGroup F] [Module π F] [m : Module π E] (B : E ββ[π] F ββ[π] π) :
+#print WeakBilin.instModule' /-
+instance (priority := 100) instModule' [CommSemiring π] [CommSemiring π] [AddCommGroup E]
+ [Module π E] [AddCommGroup F] [Module π F] [m : Module π E] (B : E ββ[π] F ββ[π] π) :
Module π (WeakBilin B) :=
m
-#align weak_bilin.module' WeakBilin.module'
+#align weak_bilin.module' WeakBilin.instModule'
+-/
instance [CommSemiring π] [CommSemiring π] [AddCommGroup E] [Module π E] [AddCommGroup F]
[Module π F] [SMul π π] [Module π E] [s : IsScalarTower π π E] (B : E ββ[π] F ββ[π] π) :
@@ -114,26 +118,56 @@ variable (B : E ββ[π] F ββ[π] π)
instance : TopologicalSpace (WeakBilin B) :=
TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
+/- warning: weak_bilin.coe_fn_continuous -> WeakBilin.coeFn_continuous is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] (B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))), Continuous.{u2, max u3 u1} (WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (F -> π) (WeakBilin.topologicalSpace.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u3, u1} F (fun (y : F) => π) (fun (a : F) => _inst_1)) (fun (x : WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (y : F) => coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} 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+but is expected to have type
+ forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 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(Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, max u2 u1} π π E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
+Case conversion may be inaccurate. Consider using '#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuousβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is continuous. -/
theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
continuous_induced_dom
#align weak_bilin.coe_fn_continuous WeakBilin.coeFn_continuous
+/- warning: weak_bilin.eval_continuous -> WeakBilin.eval_continuous is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] (B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) (y : F), Continuous.{u2, u1} (WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) π (WeakBilin.topologicalSpace.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_1 (fun (x : WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) => coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π 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_inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π 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π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) => E -> (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u1} π π E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) B x) y)
+but is expected to have type
+ forall {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u2, u1} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))))))) (y : F), Continuous.{u3, u2} (WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) y) (WeakBilin.instTopologicalSpace.{u2, u3, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_1 (fun (x : WeakBilin.{u2, u3, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) x) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y)
+Case conversion may be inaccurate. Consider using '#align weak_bilin.eval_continuous WeakBilin.eval_continuousβ'. -/
theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
(continuous_pi_iff.mp (coeFn_continuous B)) y
#align weak_bilin.eval_continuous WeakBilin.eval_continuous
+/- warning: weak_bilin.continuous_of_continuous_eval -> WeakBilin.continuous_of_continuous_eval is a dubious translation:
+lean 3 declaration is
+ forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} {F : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} π] [_inst_2 : CommSemiring.{u2} π] [_inst_3 : AddCommMonoid.{u3} E] [_inst_4 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u4} F] [_inst_6 : Module.{u2, u4} π F (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5] (B : LinearMap.{u2, u2, u3, max u4 u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) E (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π 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(CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2)))))) => E -> (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) (LinearMap.hasCoeToFun.{u2, u2, u3, max u4 u2} π π E (LinearMap.{u2, u2, u4, u2} π π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u2, u2, u4, u2} π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_4 (LinearMap.module.{u2, u2, u2, u4, u2} π π π F π (CommSemiring.toSemiring.{u2} π _inst_2) (CommSemiring.toSemiring.{u2} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B (g a)) y)) -> (Continuous.{u1, u3} Ξ± (WeakBilin.{u2, u3, u4} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_7 (WeakBilin.topologicalSpace.{u2, u3, u4} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) g)
+but is expected to have type
+ forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) [_inst_7 : TopologicalSpace.{u4} Ξ±] {g : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)}, (forall (y : F), Continuous.{u4, u3} Ξ± ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) y) _inst_7 _inst_1 (fun (a : Ξ±) => FunLike.coe.{max (succ u3) (succ u1), succ u1, succ u3} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (g a)) F (fun (_x : F) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : F) => π) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π 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(CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) 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_inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, max u3 u1} π π E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B (g a)) y)) -> (Continuous.{u4, u2} Ξ± (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) _inst_7 (WeakBilin.instTopologicalSpace.{u3, u2, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) g)
+Case conversion may be inaccurate. Consider using '#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakBilin B}
(h : β y, Continuous fun a => B (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
#align weak_bilin.continuous_of_continuous_eval WeakBilin.continuous_of_continuous_eval
+/- warning: weak_bilin.embedding -> WeakBilin.embedding is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : TopologicalSpace.{u1} π] [_inst_2 : CommSemiring.{u1} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u3} F] [_inst_6 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5] {B : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))}, (Function.Injective.{succ u2, max (succ u3) (succ u1)} E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (coeFn.{max (succ u2) (succ (max u3 u1)), max (succ u2) (succ (max u3 u1))} (LinearMap.{u1, 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(CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2)))))) => E -> (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u1} π π E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) B)) -> (Embedding.{u2, max u3 u1} (WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (F -> π) (WeakBilin.topologicalSpace.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (Pi.topologicalSpace.{u3, u1} F (fun (y : F) => π) (fun (a : F) => _inst_1)) (fun (x : WeakBilin.{u1, u2, u3} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B) (y : F) => coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} 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(Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u1} π π E (LinearMap.{u1, u1, u3, u1} π π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u3, u1} π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_4 (LinearMap.module.{u1, u1, u1, u3, u1} π π π F π (CommSemiring.toSemiring.{u1} π _inst_2) (CommSemiring.toSemiring.{u1} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2))) (CommSemiring.toSemiring.{u1} π _inst_2) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_2) (Monoid.toMulAction.{u1} π (CommMonoid.toMonoid.{u1} π (CommSemiring.toCommMonoid.{u1} π _inst_2))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_2)))) B x) y))
+but is expected to have type
+ forall {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] {B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))}, (Function.Injective.{succ u2, max (succ u3) (succ u1)} E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u2, max (succ u3) (succ u1)} (LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π 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+Case conversion may be inaccurate. Consider using '#align weak_bilin.embedding WeakBilin.embeddingβ'. -/
/-- The coercion `(Ξ» x y, B x y) : E β (F β π)` is an embedding. -/
theorem embedding {B : E ββ[π] F ββ[π] π} (hB : Function.Injective B) :
Embedding fun (x : WeakBilin B) y => B x y :=
Function.Injective.embedding_induced <| LinearMap.coe_injective.comp hB
#align weak_bilin.embedding WeakBilin.embedding
+/- warning: weak_bilin.tendsto_iff_forall_eval_tendsto -> WeakBilin.tendsto_iff_forall_eval_tendsto is a dubious translation:
+lean 3 declaration is
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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2))) (CommSemiring.toSemiring.{u2} π _inst_2) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_2) (Monoid.toMulAction.{u2} π (CommMonoid.toMonoid.{u2} π (CommSemiring.toCommMonoid.{u2} π _inst_2))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_2)))) B x) y))))
+but is expected to have type
+ forall {Ξ± : Type.{u4}} {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} π] [_inst_2 : CommSemiring.{u3} π] [_inst_3 : AddCommMonoid.{u2} E] [_inst_4 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3] [_inst_5 : AddCommMonoid.{u1} F] [_inst_6 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5] (B : LinearMap.{u3, u3, u2, max u3 u1} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))))))) {l : Filter.{u4} Ξ±} {f : Ξ± -> (WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B)} {x : WeakBilin.{u3, u2, u1} π E F _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 B}, (Function.Injective.{succ u2, max (succ u3) (succ u1)} E (LinearMap.{u3, u3, u1, u3} π π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π 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(RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) F π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_3 (LinearMap.addCommMonoid.{u3, u3, u1, u3} π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_4 (LinearMap.instModuleLinearMapAddCommMonoid.{u3, u3, u3, u1, u3} π π π F π (CommSemiring.toSemiring.{u3} π _inst_2) (CommSemiring.toSemiring.{u3} π _inst_2) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) _inst_6 (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2))) (CommSemiring.toSemiring.{u3} π _inst_2) (Semiring.toModule.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (smulCommClass_self.{u3, u3} π π (CommSemiring.toCommMonoid.{u3} π _inst_2) (MulActionWithZero.toMulAction.{u3, u3} π π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)) (CommMonoidWithZero.toZero.{u3} π (CommSemiring.toCommMonoidWithZero.{u3} π _inst_2)) (MonoidWithZero.toMulActionWithZero.{u3} π (Semiring.toMonoidWithZero.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_2)))) B x) y))))
+Case conversion may be inaccurate. Consider using '#align weak_bilin.tendsto_iff_forall_eval_tendsto WeakBilin.tendsto_iff_forall_eval_tendstoβ'. -/
theorem tendsto_iff_forall_eval_tendsto {l : Filter Ξ±} {f : Ξ± β WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (π x) β β y, Tendsto (fun i => B (f i) y) l (π (B x y)) := by
@@ -191,6 +225,12 @@ end WeakTopology
section WeakStarTopology
+/- warning: top_dual_pairing -> topDualPairing is a dubious translation:
+lean 3 declaration is
+ forall (π : Type.{u1}) (E : Type.{u2}) [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : AddCommMonoid.{u2} E] [_inst_5 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))], LinearMap.{u1, u1, max u2 u1, max u2 u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (LinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E π _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u1, u1, u2, u1} π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (ContinuousLinearMap.module.{u1, u1, u1, u2, u1} π π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_6 _inst_4 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (topDualPairing._proof_1.{u1} π _inst_1) _inst_7 (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) _inst_3) (LinearMap.module.{u1, u1, u1, u2, u1} π π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (CommSemiring.toSemiring.{u1} π _inst_1) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (topDualPairing._proof_2.{u1} π _inst_1))
+but is expected to have type
+ forall (π : Type.{u1}) (E : Type.{u2}) [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : AddCommMonoid.{u2} E] [_inst_5 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4] [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))], LinearMap.{u1, u1, max u1 u2, max u1 u2} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (LinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E π _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_6 _inst_4 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u1, u1, u2, u1} π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (ContinuousLinearMap.module.{u1, u1, u1, u2, u1} π π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_6 _inst_4 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_1) (MulActionWithZero.toMulAction.{u1, u1} π π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u1} π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))) _inst_7 (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u1, u1, u1, u2, u1} π π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_4 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_5 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (CommSemiring.toSemiring.{u1} π _inst_1) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (smulCommClass_self.{u1, u1} π π (CommSemiring.toCommMonoid.{u1} π _inst_1) (MulActionWithZero.toMulAction.{u1, u1} π π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u1} π (Semiring.toMonoidWithZero.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))
+Case conversion may be inaccurate. Consider using '#align top_dual_pairing topDualPairingβ'. -/
/-- The canonical pairing of a vector space and its topological dual. -/
def topDualPairing (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [AddCommMonoid E]
[Module π E] [TopologicalSpace E] [ContinuousConstSMul π π] : (E βL[π] π) ββ[π] E ββ[π] π :=
@@ -203,11 +243,23 @@ variable [ContinuousConstSMul π π]
variable [AddCommMonoid E] [Module π E] [TopologicalSpace E]
-theorem dual_pairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
+/- warning: dual_pairing_apply -> topDualPairing_apply is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (v : ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π 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_inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (LinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E 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(Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (LinearMap.hasCoeToFun.{u1, u1, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (LinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u1, u1, u2, u1} π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (ContinuousLinearMap.module.{u1, u1, u1, u2, u1} π π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (topDualPairing._proof_1.{u1} π _inst_1) _inst_4 (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) _inst_3) (LinearMap.module.{u1, u1, u1, u2, u1} π π π E π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (CommSemiring.toSemiring.{u1} π _inst_1) (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)) (topDualPairing._proof_2.{u1} π _inst_1)) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) (topDualPairing.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) v) x) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) => E -> π) (ContinuousLinearMap.toFun.{u1, u1, u2, u1} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) v x)
+but is expected to have type
+ forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (v : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (x : E), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => π) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) v) E (fun (_x : E) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : E) => π) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (FunLike.coe.{max (succ u2) (succ u1), max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u2, u2, max u2 u1, max u2 u1} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (fun (_x : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) v) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) v x)
+Case conversion may be inaccurate. Consider using '#align dual_pairing_apply topDualPairing_applyβ'. -/
+theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
rfl
-#align dual_pairing_apply dual_pairing_apply
+#align dual_pairing_apply topDualPairing_apply
/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
+/- warning: weak_dual -> WeakDual is a dubious translation:
+lean 3 declaration is
+ forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8))))))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{max u2 u1}
+but is expected to have type
+ forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Algebra.toSMul.{u1, u1} π π _inst_8 (CommSemiring.toSemiring.{u1} π _inst_8) (Algebra.id.{u1} π _inst_8))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{max u1 u2}
+Case conversion may be inaccurate. Consider using '#align weak_dual WeakDualβ'. -/
/-- The weak star topology is the topology coarsest topology on `E βL[π] π` such that all
functionals `Ξ» v, top_dual_pairing π E v x` are continuous. -/
def WeakDual (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [ContinuousConstSMul π π]
@@ -222,9 +274,15 @@ namespace WeakDual
instance : Inhabited (WeakDual π E) :=
ContinuousLinearMap.inhabited
-instance WeakDual.continuousLinearMapClass : ContinuousLinearMapClass (WeakDual π E) π E π :=
+/- warning: weak_dual.weak_dual.continuous_linear_map_class -> WeakDual.instContinuousLinearMapClass is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E], ContinuousLinearMapClass.{max u2 u1, u1, u2, u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))
+but is expected to have type
+ forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E], ContinuousLinearMapClass.{max u2 u1, u1, u2, u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π (CommSemiring.toSemiring.{u1} π _inst_1) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_6 (Semiring.toModule.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))
+Case conversion may be inaccurate. Consider using '#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.instContinuousLinearMapClassβ'. -/
+instance WeakDual.instContinuousLinearMapClass : ContinuousLinearMapClass (WeakDual π E) π E π :=
ContinuousLinearMap.continuousSemilinearMapClass
-#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.WeakDual.continuousLinearMapClass
+#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.instContinuousLinearMapClass
/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`
directly. -/
@@ -243,12 +301,18 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [C
DistribMulAction M (WeakDual π E) :=
ContinuousLinearMap.distribMulAction
+/- warning: weak_dual.module' -> WeakDual.instModule' is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (R : Type.{u3}) [_inst_8 : Semiring.{u3} R] [_inst_9 : Module.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))] [_inst_10 : SMulCommClass.{u1, u3, u1} π R π (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))) (SMulZeroClass.toHasSmul.{u3, u1} R π (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (SMulWithZero.toSmulZeroClass.{u3, u1} R π (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)))) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))] [_inst_11 : ContinuousConstSMul.{u3, u1} R π _inst_2 (SMulZeroClass.toHasSmul.{u3, u1} R π (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (SMulWithZero.toSmulZeroClass.{u3, u1} R π (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)))) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (AddZeroClass.toHasZero.{u1} π (AddMonoid.toAddZeroClass.{u1} π (AddCommMonoid.toAddMonoid.{u1} π (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))], Module.{u3, max u2 u1} R (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)
+but is expected to have type
+ forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (R : Type.{u3}) [_inst_8 : Semiring.{u3} R] [_inst_9 : Module.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))] [_inst_10 : SMulCommClass.{u1, u3, u1} π R π (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1)) (SMulZeroClass.toSMul.{u3, u1} R π (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (SMulWithZero.toSMulZeroClass.{u3, u1} R π (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))] [_inst_11 : ContinuousConstSMul.{u3, u1} R π _inst_2 (SMulZeroClass.toSMul.{u3, u1} R π (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (SMulWithZero.toSMulZeroClass.{u3, u1} R π (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_8)) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (MulActionWithZero.toSMulWithZero.{u3, u1} R π (Semiring.toMonoidWithZero.{u3} R _inst_8) (CommMonoidWithZero.toZero.{u1} π (CommSemiring.toCommMonoidWithZero.{u1} π _inst_1)) (Module.toMulActionWithZero.{u3, u1} R π _inst_8 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))) _inst_9))))], Module.{u3, max u2 u1} R (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.instAddCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)
+Case conversion may be inaccurate. Consider using '#align weak_dual.module' WeakDual.instModule'β'. -/
/-- If `π` is a topological module over a semiring `R` and scalar multiplication commutes with the
multiplication on `π`, then `weak_dual π E` is a module over `R`. -/
-instance module' (R) [Semiring R] [Module R π] [SMulCommClass π R π] [ContinuousConstSMul R π] :
+instance instModule' (R) [Semiring R] [Module R π] [SMulCommClass π R π] [ContinuousConstSMul R π] :
Module R (WeakDual π E) :=
ContinuousLinearMap.module
-#align weak_dual.module' WeakDual.module'
+#align weak_dual.module' WeakDual.instModule'
instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [ContinuousConstSMul M π] :
ContinuousConstSMul M (WeakDual π E) :=
@@ -262,14 +326,32 @@ instance (M) [Monoid M] [DistribMulAction M π] [SMulCommClass π M π] [T
β¨continuous_induced_rng.2 <|
continuous_fst.smul ((WeakBilin.coeFn_continuous (topDualPairing π E)).comp continuous_snd)β©
+/- warning: weak_dual.coe_fn_continuous -> WeakDual.coeFn_continuous is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E], Continuous.{max u2 u1, max u2 u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (E -> π) (WeakDual.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Pi.topologicalSpace.{u2, u1} E (fun (y : E) => π) (fun (a : E) => _inst_2)) (fun (x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (y : E) => coeFn.{succ (max u2 u1), max (succ u2) (succ u1)} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) x y)
+but is expected to have type
+ forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E], Continuous.{max u2 u1, max u2 u1} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (forall (y : E), (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Pi.topologicalSpace.{u1, u2} E (fun (y : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) (fun (a : E) => _inst_2)) (fun (x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (y : E) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) x y)
+Case conversion may be inaccurate. Consider using '#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuousβ'. -/
theorem coeFn_continuous : Continuous fun (x : WeakDual π E) y => x y :=
continuous_induced_dom
#align weak_dual.coe_fn_continuous WeakDual.coeFn_continuous
+/- warning: weak_dual.eval_continuous -> WeakDual.eval_continuous is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] (y : E), Continuous.{max u2 u1, u1} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π (WeakDual.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_2 (fun (x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => coeFn.{succ (max u2 u1), max (succ u2) (succ u1)} (WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) x y)
+but is expected to have type
+ forall {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] (y : E), Continuous.{max u2 u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) ((fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_2 (fun (x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) x y)
+Case conversion may be inaccurate. Consider using '#align weak_dual.eval_continuous WeakDual.eval_continuousβ'. -/
theorem eval_continuous (y : E) : Continuous fun x : WeakDual π E => x y :=
continuous_pi_iff.mp coeFn_continuous y
#align weak_dual.eval_continuous WeakDual.eval_continuous
+/- warning: weak_dual.continuous_of_continuous_eval -> WeakDual.continuous_of_continuous_eval is a dubious translation:
+lean 3 declaration is
+ forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toHasAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Mul.toSMul.{u2} π (Distrib.toHasMul.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u3} E] [_inst_6 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u3} E] [_inst_8 : TopologicalSpace.{u1} Ξ±] {g : Ξ± -> (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)}, (forall (y : E), Continuous.{u1, u2} Ξ± π _inst_8 _inst_2 (fun (a : Ξ±) => coeFn.{succ (max u3 u2), max (succ u3) (succ u2)} (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => E -> π) (WeakDual.hasCoeToFun.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (g a) y)) -> (Continuous.{u1, max u3 u2} Ξ± (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.topologicalSpace.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) g)
+but is expected to have type
+ forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] [_inst_8 : TopologicalSpace.{u3} Ξ±] {g : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)}, (forall (y : E), Continuous.{u3, u2} Ξ± ((fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) y) _inst_8 _inst_2 (fun (a : Ξ±) => FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E (fun (_x : E) => (fun (a._@.Mathlib.Topology.Algebra.Module.WeakDual._hyg.1928 : E) => π) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) E π _inst_7 _inst_2 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (WeakDual.instContinuousLinearMapClass.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7))) (g a) y)) -> (Continuous.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_8 (WeakDual.instTopologicalSpace.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) g)
+Case conversion may be inaccurate. Consider using '#align weak_dual.continuous_of_continuous_eval WeakDual.continuous_of_continuous_evalβ'. -/
theorem continuous_of_continuous_eval [TopologicalSpace Ξ±] {g : Ξ± β WeakDual π E}
(h : β y, Continuous fun a => (g a) y) : Continuous g :=
continuous_induced_rng.2 (continuous_pi_iff.mpr h)
@@ -283,6 +365,12 @@ instance [T2Space π] : T2Space (WeakDual π E) :=
end WeakDual
/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler module[module] π -/
+/- warning: weak_space -> WeakSpace is a dubious translation:
+lean 3 declaration is
+ forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8))))))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{u2}
+but is expected to have type
+ forall (π : Type.{u1}) (E : Type.{u2}) [_inst_8 : CommSemiring.{u1} π] [_inst_9 : TopologicalSpace.{u1} π] [_inst_10 : ContinuousAdd.{u1} π _inst_9 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_8)))))] [_inst_11 : ContinuousConstSMul.{u1, u1} π π _inst_9 (Algebra.toSMul.{u1, u1} π π _inst_8 (CommSemiring.toSemiring.{u1} π _inst_8) (Algebra.id.{u1} π _inst_8))] [_inst_12 : AddCommMonoid.{u2} E] [_inst_13 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_8) _inst_12] [_inst_14 : TopologicalSpace.{u2} E], Type.{u2}
+Case conversion may be inaccurate. Consider using '#align weak_space WeakSpaceβ'. -/
/-- The weak topology is the topology coarsest topology on `E` such that all
functionals `Ξ» x, top_dual_pairing π E v x` are continuous. -/
@[nolint has_nonempty_instance]
@@ -297,6 +385,12 @@ namespace WeakSpace
variable {π E F} [AddCommMonoid F] [Module π F] [TopologicalSpace F]
+/- warning: weak_space.map -> WeakSpace.map is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u3} F] [_inst_9 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u3} F], (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) -> (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10))
+but is expected to have type
+ forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Algebra.toSMul.{u1, u1} π π _inst_1 (CommSemiring.toSemiring.{u1} π _inst_1) (Algebra.id.{u1} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u3} F] [_inst_9 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u3} F], (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) -> (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10))
+Case conversion may be inaccurate. Consider using '#align weak_space.map WeakSpace.mapβ'. -/
/-- A continuous linear map from `E` to `F` is still continuous when `E` and `F` are equipped with
their weak topologies. -/
def map (f : E βL[π] F) : WeakSpace π E βL[π] WeakSpace π F :=
@@ -305,10 +399,22 @@ def map (f : E βL[π] F) : WeakSpace π E βL[π] WeakSpace π F :=
WeakBilin.continuous_of_continuous_eval _ fun l => WeakBilin.eval_continuous _ (l βL f) }
#align weak_space.map WeakSpace.map
+/- warning: weak_space.map_apply -> WeakSpace.map_apply is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u3} F] [_inst_9 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u3} F] (f : ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) (x : E), Eq.{succ u3} (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) => (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) -> (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (WeakSpace.map.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 _inst_9 _inst_10 f) x) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) => E -> F) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) f x)
+but is expected to have type
+ forall {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : CommSemiring.{u3} π] [_inst_2 : TopologicalSpace.{u3} π] [_inst_3 : ContinuousAdd.{u3} π _inst_2 (Distrib.toAdd.{u3} π (NonUnitalNonAssocSemiring.toDistrib.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u3, u3} π π _inst_2 (Algebra.toSMul.{u3, u3} π π _inst_1 (CommSemiring.toSemiring.{u3} π _inst_1) (Algebra.id.{u3} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u1} F] [_inst_9 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u1} F] (f : ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) (x : E), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π 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(WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)))) (WeakSpace.map.{u3, u2, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 _inst_9 _inst_10 f) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) E F _inst_7 _inst_10 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9))) f x)
+Case conversion may be inaccurate. Consider using '#align weak_space.map_apply WeakSpace.map_applyβ'. -/
theorem map_apply (f : E βL[π] F) (x : E) : WeakSpace.map f x = f x :=
rfl
#align weak_space.map_apply WeakSpace.map_apply
+/- warning: weak_space.coe_map -> WeakSpace.coe_map is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} {E : Type.{u2}} {F : Type.{u3}} [_inst_1 : CommSemiring.{u1} π] [_inst_2 : TopologicalSpace.{u1} π] [_inst_3 : ContinuousAdd.{u1} π _inst_2 (Distrib.toHasAdd.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u1, u1} π π _inst_2 (Mul.toSMul.{u1} π (Distrib.toHasMul.{u1} π (NonUnitalNonAssocSemiring.toDistrib.{u1} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u1, u2} π E (CommSemiring.toSemiring.{u1} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u3} F] [_inst_9 : Module.{u1, u3} π F (CommSemiring.toSemiring.{u1} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u3} F] (f : ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9), Eq.{max (succ u2) (succ u3)} ((fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) => (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) -> (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (WeakSpace.map.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 _inst_9 _inst_10 f)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) => (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) -> (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) (WeakSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.topologicalSpace.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.addCommMonoid.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.topologicalSpace.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.addCommMonoid.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.module.{u1, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.module.{u1, u3} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (WeakSpace.map.{u1, u2, u3} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 _inst_9 _inst_10 f)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) => E -> F) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} π π (CommSemiring.toSemiring.{u1} π _inst_1) (CommSemiring.toSemiring.{u1} π _inst_1) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (CommSemiring.toSemiring.{u1} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) f)
+but is expected to have type
+ forall {π : Type.{u3}} {E : Type.{u2}} {F : Type.{u1}} [_inst_1 : CommSemiring.{u3} π] [_inst_2 : TopologicalSpace.{u3} π] [_inst_3 : ContinuousAdd.{u3} π _inst_2 (Distrib.toAdd.{u3} π (NonUnitalNonAssocSemiring.toDistrib.{u3} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u3, u3} π π _inst_2 (Algebra.toSMul.{u3, u3} π π _inst_1 (CommSemiring.toSemiring.{u3} π _inst_1) (Algebra.id.{u3} π _inst_1))] [_inst_5 : AddCommMonoid.{u2} E] [_inst_6 : Module.{u3, u2} π E (CommSemiring.toSemiring.{u3} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u2} E] [_inst_8 : AddCommMonoid.{u1} F] [_inst_9 : Module.{u3, u1} π F (CommSemiring.toSemiring.{u3} π _inst_1) _inst_8] [_inst_10 : TopologicalSpace.{u1} F] (f : ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9), Eq.{max (succ u2) (succ u1)} (forall (a : WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (fun (_x : WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) => WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) (WeakSpace.{u3, u2} π E 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_inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)) π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) (WeakSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instTopologicalSpace.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instAddCommMonoid.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instTopologicalSpace.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instAddCommMonoid.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10) (WeakSpace.instModule.{u3, u2} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (WeakSpace.instModule.{u3, u1} π F _inst_1 _inst_2 _inst_3 _inst_4 _inst_8 _inst_9 _inst_10)))) (WeakSpace.map.{u3, u2, u1} π E F _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 _inst_9 _inst_10 f)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) E (fun (_x : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) E F _inst_7 _inst_10 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9) π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} π π (CommSemiring.toSemiring.{u3} π _inst_1) (CommSemiring.toSemiring.{u3} π _inst_1) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (CommSemiring.toSemiring.{u3} π _inst_1))) E _inst_7 _inst_5 F _inst_10 _inst_8 _inst_6 _inst_9))) f)
+Case conversion may be inaccurate. Consider using '#align weak_space.coe_map WeakSpace.coe_mapβ'. -/
@[simp]
theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
rfl
@@ -316,6 +422,12 @@ theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
end WeakSpace
+/- warning: tendsto_iff_forall_eval_tendsto_top_dual_pairing -> tendsto_iff_forall_eval_tendsto_topDualPairing is a dubious translation:
+lean 3 declaration is
+ forall {Ξ± : Type.{u1}} {π : Type.{u2}} {E : Type.{u3}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toHasAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Mul.toSMul.{u2} π (Distrib.toHasMul.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))] [_inst_5 : AddCommMonoid.{u3} E] [_inst_6 : Module.{u2, u3} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u3} E] {l : Filter.{u1} Ξ±} {f : Ξ± -> (WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u3} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff 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+but is expected to have type
+ forall {Ξ± : Type.{u3}} {π : Type.{u2}} {E : Type.{u1}} [_inst_1 : CommSemiring.{u2} π] [_inst_2 : TopologicalSpace.{u2} π] [_inst_3 : ContinuousAdd.{u2} π _inst_2 (Distrib.toAdd.{u2} π (NonUnitalNonAssocSemiring.toDistrib.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))] [_inst_4 : ContinuousConstSMul.{u2, u2} π π _inst_2 (Algebra.toSMul.{u2, u2} π π _inst_1 (CommSemiring.toSemiring.{u2} π _inst_1) (Algebra.id.{u2} π _inst_1))] [_inst_5 : AddCommMonoid.{u1} E] [_inst_6 : Module.{u2, u1} π E (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5] [_inst_7 : TopologicalSpace.{u1} E] {l : Filter.{u3} Ξ±} {f : Ξ± -> (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)} {x : WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7}, Iff (Filter.Tendsto.{u3, max u2 u1} Ξ± (WeakDual.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) f l 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))))) (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (fun (_x : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) => LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u2 u1, max u2 u1} π π (ContinuousLinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (LinearMap.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E π _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (ContinuousLinearMap.addCommMonoid.{u2, u2, u1, u2} π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) E _inst_7 _inst_5 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) _inst_3) (LinearMap.addCommMonoid.{u2, u2, u1, u2} π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (ContinuousLinearMap.module.{u2, u2, u2, u1, u2} π π π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) E _inst_7 _inst_5 _inst_6 π _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))))) _inst_4 (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) _inst_3) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} π π π E π (CommSemiring.toSemiring.{u2} π _inst_1) (CommSemiring.toSemiring.{u2} π _inst_1) _inst_5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} π (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) _inst_6 (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1))) (CommSemiring.toSemiring.{u2} π _inst_1) (Semiring.toModule.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (smulCommClass_self.{u2, u2} π π (CommSemiring.toCommMonoid.{u2} π _inst_1) (MulActionWithZero.toMulAction.{u2, u2} π π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)) (CommMonoidWithZero.toZero.{u2} π (CommSemiring.toCommMonoidWithZero.{u2} π _inst_1)) (MonoidWithZero.toMulActionWithZero.{u2} π (Semiring.toMonoidWithZero.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (CommSemiring.toSemiring.{u2} π _inst_1)))) (topDualPairing.{u2, u1} π E _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_4) x) y)))
+Case conversion may be inaccurate. Consider using '#align tendsto_iff_forall_eval_tendsto_top_dual_pairing tendsto_iff_forall_eval_tendsto_topDualPairingβ'. -/
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter Ξ±} {f : Ξ± β WeakDual π E}
{x : WeakDual π E} :
Tendsto f l (π x) β
mathlib commit https://github.com/leanprover-community/mathlib/commit/3cacc945118c8c637d89950af01da78307f59325
@@ -194,7 +194,7 @@ section WeakStarTopology
/-- The canonical pairing of a vector space and its topological dual. -/
def topDualPairing (π E) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π] [AddCommMonoid E]
[Module π E] [TopologicalSpace E] [ContinuousConstSMul π π] : (E βL[π] π) ββ[π] E ββ[π] π :=
- ContinuousLinearMap.coeLm π
+ ContinuousLinearMap.coeLM π
#align top_dual_pairing topDualPairing
variable [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π]
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
(This PR continues #11472)
Prove the following basic facts about the weak topology in a topological vector space (or AddCommMonoid) E
.
E
that converges to a point in the original topology (with respect to some filter) converges also in the weak topology.E
converges to a point in the original topology, then its composition with a continuous linear functional converges.E
that is continuous in the weak topology is also continuous in the original topology.Motivation: WeakSpace has some basic properties, analogue of which are proved for WeakDual.
I have some doubts:
continuousLinearMapToWeakSpace : E βL[π] WeakSpace π E
when needed. Is this fine or is there a canonical way to avoid it? (I tried to add variable (π E) in
before def/thm but it gave type mismatch in many places)@@ -328,6 +328,39 @@ theorem coe_map (f : E βL[π] F) : (WeakSpace.map f : E β F) = f :=
end WeakSpace
+variable (π E) in
+/-- There is a canonical map `E β WeakSpace π E` (the "identity"
+mapping). It is a linear equivalence. -/
+def toWeakSpace : E ββ[π] WeakSpace π E := LinearEquiv.refl π E
+
+variable (π E) in
+/-- For a topological vector space `E`, "identity mapping" `E β WeakSpace π E` is continuous.
+This definition implements it as a continuous linear map. -/
+def continuousLinearMapToWeakSpace : E βL[π] WeakSpace π E where
+ __ := toWeakSpace π E
+ cont := by
+ apply WeakBilin.continuous_of_continuous_eval
+ exact ContinuousLinearMap.continuous
+
+variable (π E) in
+@[simp]
+theorem continuousLinearMapToWeakSpace_eq_toWeakSpace (x : E) :
+ continuousLinearMapToWeakSpace π E x = toWeakSpace π E x := by rfl
+
+theorem continuousLinearMapToWeakSpace_bijective :
+ Function.Bijective (continuousLinearMapToWeakSpace π E) :=
+ (toWeakSpace π E).bijective
+
+/-- The canonical map from `WeakSpace π E` to `E` is an open map. -/
+theorem isOpenMap_toWeakSpace_symm : IsOpenMap (toWeakSpace π E).symm :=
+ IsOpenMap.of_inverse (continuousLinearMapToWeakSpace π E).cont
+ (toWeakSpace π E).left_inv (toWeakSpace π E).right_inv
+
+/-- A set in `E` which is open in the weak topology is open. -/
+theorem WeakSpace.isOpen_of_isOpen (V : Set E)
+ (hV : IsOpen ((continuousLinearMapToWeakSpace π E) '' V : Set (WeakSpace π E))) : IsOpen V := by
+ simpa [Set.image_image] using isOpenMap_toWeakSpace_symm _ hV
+
theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter Ξ±} {f : Ξ± β WeakDual π E}
{x : WeakDual π E} :
Tendsto f l (π x) β
Empty lines were removed by executing the following Python script twice
import os
import re
# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
for filename in files:
if filename.endswith('.lean'):
file_path = os.path.join(dir_path, filename)
# Open the file and read its contents
with open(file_path, 'r') as file:
content = file.read()
# Use a regular expression to replace sequences of "variable" lines separated by empty lines
# with sequences without empty lines
modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)
# Write the modified content back to the file
with open(file_path, 'w') as file:
file.write(modified_content)
@@ -103,11 +103,8 @@ instance instIsScalarTower [CommSemiring π] [CommSemiring π] [AddCommGroup
section Semiring
variable [TopologicalSpace π] [CommSemiring π]
-
variable [AddCommMonoid E] [Module π E]
-
variable [AddCommMonoid F] [Module π F]
-
variable (B : E ββ[π] F ββ[π] π)
instance instTopologicalSpace : TopologicalSpace (WeakBilin B) :=
@@ -162,9 +159,7 @@ end Semiring
section Ring
variable [TopologicalSpace π] [CommRing π]
-
variable [AddCommGroup E] [Module π E]
-
variable [AddCommGroup F] [Module π F]
@@ -195,9 +190,7 @@ def topDualPairing (π E) [CommSemiring π] [TopologicalSpace π] [Continu
#align top_dual_pairing topDualPairing
variable [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π]
-
variable [ContinuousConstSMul π π]
-
variable [AddCommMonoid E] [Module π E] [TopologicalSpace E]
theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing π E v x = v x :=
The FunLike hierarchy is very big and gets scanned through each time we need a coercion (via the CoeFun
instance). It looks like unbundled inheritance suits Lean 4 better here. The only class that still extends FunLike
is EquivLike
, since that has a custom coe_injective'
field that is easier to implement. All other classes should take FunLike
or EquivLike
as a parameter.
Previously, morphism classes would be Type
-valued and extend FunLike
:
/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
extends FunLike F A B :=
(map_op : β (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))
After this PR, they should be Prop
-valued and take FunLike
as a parameter:
/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
[FunLike F A B] : Prop :=
(map_op : β (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))
(Note that A B
stay marked as outParam
even though they are not purely required to be so due to the FunLike
parameter already filling them in. This is required to see through type synonyms, which is important in the category theory library. Also, I think keeping them as outParam
is slightly faster.)
Similarly, MyEquivClass
should take EquivLike
as a parameter.
As a result, every mention of [MyHomClass F A B]
should become [FunLike F A B] [MyHomClass F A B]
.
While overall this gives some great speedups, there are some cases that are noticeably slower. In particular, a failing application of a lemma such as map_mul
is more expensive. This is due to suboptimal processing of arguments. For example:
variable [FunLike F M N] [Mul M] [Mul N] (f : F) (x : M) (y : M)
theorem map_mul [MulHomClass F M N] : f (x * y) = f x * f y
example [AddHomClass F A B] : f (x * y) = f x * f y := map_mul f _ _
Before this PR, applying map_mul f
gives the goals [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]
. Since M
and N
are out_param
s, [MulHomClass F ?M ?N]
is synthesized first, supplies values for ?M
and ?N
and then the Mul M
and Mul N
instances can be found.
After this PR, the goals become [FunLike F ?M ?N] [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]
. Now [FunLike F ?M ?N]
is synthesized first, supplies values for ?M
and ?N
and then the Mul M
and Mul N
instances can be found, before trying MulHomClass F M N
which fails. Since the Mul
hierarchy is very big, this can be slow to fail, especially when there is no such Mul
instance.
A long-term but harder to achieve solution would be to specify the order in which instance goals get solved. For example, we'd like to change the arguments to map_mul
to look like [FunLike F M N] [Mul M] [Mul N] [highPriority <| MulHomClass F M N]
because MulHomClass
fails or succeeds much faster than the others.
As a consequence, the simpNF
linter is much slower since by design it tries and fails to apply many map_
lemmas. The same issue occurs a few times in existing calls to simp [map_mul]
, where map_mul
is tried "too soon" and fails. Thanks to the speedup of leanprover/lean4#2478 the impact is very limited, only in files that already were close to the timeout.
simp
not firing sometimesThis affects map_smulββ
and related definitions. For simp
lemmas Lean apparently uses a slightly different mechanism to find instances, so that rw
can find every argument to map_smulββ
successfully but simp
can't: leanprover/lean4#3701.
Especially in the category theory library, we might sometimes have a type A
which is also accessible as a synonym (Bundled A hA).1
. Instance synthesis doesn't always work if we have f : A β* B
but x * y : (Bundled A hA).1
or vice versa. This seems to be mostly fixed by keeping A B
as outParam
s in MulHomClass F A B
. (Presumably because Lean will do a definitional check A =?= (Bundled A hA).1
instead of using the syntax in the discrimination tree.)
The timeouts can be worked around for now by specifying which map_mul
we mean, either as map_mul f
for some explicit f
, or as e.g. MonoidHomClass.map_mul
.
map_smulββ
not firing as simp
lemma can be worked around by going back to the pre-FunLike situation and making LinearMap.map_smulββ
a simp
lemma instead of the generic map_smulββ
. Writing simp [map_smulββ _]
also works.
Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
@@ -179,10 +179,6 @@ instance instTopologicalAddGroup [ContinuousAdd π] : TopologicalAddGroup (Wea
refine' cast (congr_arg _ _) (eval_continuous B (-y))
ext x
simp only [map_neg, Function.comp_apply, LinearMap.neg_apply]
- -- Porting note: mathlib3 proof was done here
- rw [β (B x).neg_apply]
- congr
- exact (map_neg B x).symm
end Ring
@@ -233,6 +229,9 @@ instance instContinuousAdd : ContinuousAdd (WeakDual π E) :=
instance instInhabited : Inhabited (WeakDual π E) :=
ContinuousLinearMap.inhabited
+instance instFunLike : FunLike (WeakDual π E) E π :=
+ ContinuousLinearMap.funLike
+
instance instContinuousLinearMapClass : ContinuousLinearMapClass (WeakDual π E) π E π :=
ContinuousLinearMap.continuousSemilinearMapClass
#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.instContinuousLinearMapClass
FunLike
to DFunLike
(#9785)
This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.
This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:
sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>
@@ -237,10 +237,10 @@ instance instContinuousLinearMapClass : ContinuousLinearMapClass (WeakDual π
ContinuousLinearMap.continuousSemilinearMapClass
#align weak_dual.weak_dual.continuous_linear_map_class WeakDual.instContinuousLinearMapClass
-/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`
+/-- Helper instance for when there's too many metavariables to apply `DFunLike.hasCoeToFun`
directly. -/
instance : CoeFun (WeakDual π E) fun _ => E β π :=
- FunLike.hasCoeToFun
+ DFunLike.hasCoeToFun
/-- If a monoid `M` distributively continuously acts on `π` and this action commutes with
multiplication on `π`, then it acts on `WeakDual π E`. -/
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -69,7 +69,7 @@ open Filter
open Topology
-variable {Ξ± π π R E F M : Type _}
+variable {Ξ± π π R E F M : Type*}
section WeakTopology
@@ -210,7 +210,7 @@ theorem topDualPairing_apply (v : E βL[π] π) (x : E) : topDualPairing
/-- The weak star topology is the topology coarsest topology on `E βL[π] π` such that all
functionals `fun v => v x` are continuous. -/
-def WeakDual (π E : Type _) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π]
+def WeakDual (π E : Type*) [CommSemiring π] [TopologicalSpace π] [ContinuousAdd π]
[ContinuousConstSMul π π] [AddCommMonoid E] [Module π E] [TopologicalSpace E] :=
WeakBilin (topDualPairing π E)
#align weak_dual WeakDual
@@ -81,7 +81,7 @@ def WeakBilin [CommSemiring π] [AddCommMonoid E] [Module π E] [AddCommMono
namespace WeakBilin
--- Porting note: the next two instance should be derived from the definition
+-- Porting note: the next two instances should be derived from the definition
instance instAddCommMonoid [CommSemiring π] [a : AddCommMonoid E] [Module π E] [AddCommMonoid F]
[Module π F] (B : E ββ[π] F ββ[π] π) : AddCommMonoid (WeakBilin B) := a
@@ -2,15 +2,12 @@
Copyright (c) 2021 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€, Moritz Doll
-
-! This file was ported from Lean 3 source module topology.algebra.module.weak_dual
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.BilinearMap
+#align_import topology.algebra.module.weak_dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
/-!
# Weak dual topology
Now that leanprover/lean4#2210 has been merged, this PR:
set_option synthInstance.etaExperiment true
commands (and some etaExperiment%
term elaborators)set_option maxHeartbeats
commandsCo-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Matthew Ballard <matt@mrb.email>
@@ -170,7 +170,6 @@ variable [AddCommGroup E] [Module π E]
variable [AddCommGroup F] [Module π F]
-set_option synthInstance.etaExperiment true
variable (B : E ββ[π] F ββ[π] π)
The unported dependencies are