analysis.normed_space.dual ⟷ Mathlib.Analysis.NormedSpace.Dual

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 -/
 import Analysis.NormedSpace.HahnBanach.Extension
-import Analysis.NormedSpace.IsROrC
+import Analysis.NormedSpace.RCLike
 import Analysis.LocallyConvex.Polar
 
 #align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
@@ -137,7 +137,7 @@ end General
 
 section BidualIsometry
 
-variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 #print NormedSpace.norm_le_dual_bound /-
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
@@ -302,7 +302,7 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
 #print NormedSpace.polar_closedBall /-
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
 inverse radius. -/
-theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
+theorem polar_closedBall {𝕜 E : Type _} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
     (hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ :=
   by
   refine' subset.antisymm _ (closed_ball_inv_subset_polar_closed_ball _)

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -265,7 +265,7 @@ theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z
     apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
   have cancel : ‖c⁻¹‖ * ‖c‖ = 1 := by
     simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
-  rwa [cancel] at le 
+  rwa [cancel] at le
 #align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
 -/
 
@@ -274,7 +274,7 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
     polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) :=
   by
   intro x' hx'
-  rw [mem_polar_iff] at hx' 
+  rw [mem_polar_iff] at hx'
   simp only [polar, mem_set_of_eq, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
   have hcr : 0 < ‖c‖ / r := div_pos (zero_lt_one.trans hc) hr
   refine' ContinuousLinearMap.opNorm_le_of_shell hr hcr.le hc fun x h₁ h₂ => _

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -95,7 +95,7 @@ theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f =
 #print NormedSpace.inclusionInDoubleDual_norm_eq /-
 theorem inclusionInDoubleDual_norm_eq :
     ‖inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ :=
-  ContinuousLinearMap.op_norm_flip _
+  ContinuousLinearMap.opNorm_flip _
 #align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
 -/
 
@@ -107,7 +107,7 @@ theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1
 
 #print NormedSpace.double_dual_bound /-
 theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by
-  simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x
+  simpa using ContinuousLinearMap.le_of_opNorm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x
 #align normed_space.double_dual_bound NormedSpace.double_dual_bound
 -/
 
@@ -277,7 +277,7 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
   rw [mem_polar_iff] at hx' 
   simp only [polar, mem_set_of_eq, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
   have hcr : 0 < ‖c‖ / r := div_pos (zero_lt_one.trans hc) hr
-  refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x h₁ h₂ => _
+  refine' ContinuousLinearMap.opNorm_le_of_shell hr hcr.le hc fun x h₁ h₂ => _
   calc
     ‖x' x‖ ≤ 1 := hx' _ h₂
     _ ≤ ‖c‖ / r * ‖x‖ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
@@ -308,7 +308,7 @@ theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E]
   refine' subset.antisymm _ (closed_ball_inv_subset_polar_closed_ball _)
   intro x' h
   simp only [mem_closedBall_zero_iff]
-  refine' ContinuousLinearMap.op_norm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z hz => _
+  refine' ContinuousLinearMap.opNorm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z hz => _
   simpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.to_linear_map h z
 #align normed_space.polar_closed_ball NormedSpace.polar_closedBall
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -143,7 +143,16 @@ variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [Norm
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
     Compare `continuous_linear_map.op_norm_le_bound`. -/
 theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
-    ‖x‖ ≤ M := by classical
+    ‖x‖ ≤ M := by
+  classical
+  by_cases h : x = 0
+  · simp only [h, hMp, norm_zero]
+  · obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h
+    calc
+      ‖x‖ = ‖(‖x‖ : 𝕜)‖ := is_R_or_C.norm_coe_norm.symm
+      _ = ‖f x‖ := by rw [hfx]
+      _ ≤ M * ‖f‖ := (hM f)
+      _ = M := by rw [hf₁, mul_one]
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -143,16 +143,7 @@ variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [Norm
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
     Compare `continuous_linear_map.op_norm_le_bound`. -/
 theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
-    ‖x‖ ≤ M := by
-  classical
-  by_cases h : x = 0
-  · simp only [h, hMp, norm_zero]
-  · obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h
-    calc
-      ‖x‖ = ‖(‖x‖ : 𝕜)‖ := is_R_or_C.norm_coe_norm.symm
-      _ = ‖f x‖ := by rw [hfx]
-      _ ≤ M * ‖f‖ := (hM f)
-      _ = M := by rw [hf₁, mul_one]
+    ‖x‖ ≤ M := by classical
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2020 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 -/
-import Mathbin.Analysis.NormedSpace.HahnBanach.Extension
-import Mathbin.Analysis.NormedSpace.IsROrC
-import Mathbin.Analysis.LocallyConvex.Polar
+import Analysis.NormedSpace.HahnBanach.Extension
+import Analysis.NormedSpace.IsROrC
+import Analysis.LocallyConvex.Polar
 
 #align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -313,10 +313,11 @@ theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E]
 #align normed_space.polar_closed_ball NormedSpace.polar_closedBall
 -/
 
-#print NormedSpace.bounded_polar_of_mem_nhds_zero /-
+#print NormedSpace.isBounded_polar_of_mem_nhds_zero /-
 /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
 of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
-theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) : Bounded (polar 𝕜 s) :=
+theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
+    Bounded (polar 𝕜 s) :=
   by
   obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := NormedField.exists_one_lt_norm 𝕜
   obtain ⟨r, r_pos, r_ball⟩ : ∃ (r : ℝ) (hr : 0 < r), ball 0 r ⊆ s := Metric.mem_nhds_iff.1 s_nhd
@@ -324,7 +325,7 @@ theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E))
     bounded_closed_ball.mono
       (((dual_pairing 𝕜 E).flip.polar_antitone r_ball).trans <|
         polar_ball_subset_closed_ball_div ha r_pos)
-#align normed_space.bounded_polar_of_mem_nhds_zero NormedSpace.bounded_polar_of_mem_nhds_zero
+#align normed_space.bounded_polar_of_mem_nhds_zero NormedSpace.isBounded_polar_of_mem_nhds_zero
 -/
 
 end PolarSets

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
-
-! This file was ported from Lean 3 source module analysis.normed_space.dual
-! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.NormedSpace.HahnBanach.Extension
 import Mathbin.Analysis.NormedSpace.IsROrC
 import Mathbin.Analysis.LocallyConvex.Polar
 
+#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
+
 /-!
 # The topological dual of a normed space
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -88,23 +88,31 @@ def inclusionInDoubleDual : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
 #align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
 -/
 
+#print NormedSpace.dual_def /-
 @[simp]
 theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f = f x :=
   rfl
 #align normed_space.dual_def NormedSpace.dual_def
+-/
 
+#print NormedSpace.inclusionInDoubleDual_norm_eq /-
 theorem inclusionInDoubleDual_norm_eq :
     ‖inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ :=
   ContinuousLinearMap.op_norm_flip _
 #align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
+-/
 
+#print NormedSpace.inclusionInDoubleDual_norm_le /-
 theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by
   rw [inclusion_in_double_dual_norm_eq]; exact ContinuousLinearMap.norm_id_le
 #align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
+-/
 
+#print NormedSpace.double_dual_bound /-
 theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by
   simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x
 #align normed_space.double_dual_bound NormedSpace.double_dual_bound
+-/
 
 #print NormedSpace.dualPairing /-
 /-- The dual pairing as a bilinear form. -/
@@ -113,16 +121,20 @@ def dualPairing : Dual 𝕜 E →ₗ[𝕜] E →ₗ[𝕜] 𝕜 :=
 #align normed_space.dual_pairing NormedSpace.dualPairing
 -/
 
+#print NormedSpace.dualPairing_apply /-
 @[simp]
 theorem dualPairing_apply {v : Dual 𝕜 E} {x : E} : dualPairing 𝕜 E v x = v x :=
   rfl
 #align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
+-/
 
+#print NormedSpace.dualPairing_separatingLeft /-
 theorem dualPairing_separatingLeft : (dualPairing 𝕜 E).SeparatingLeft :=
   by
   rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
   exact ContinuousLinearMap.coe_injective
 #align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
+-/
 
 end General
 
@@ -130,6 +142,7 @@ section BidualIsometry
 
 variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
+#print NormedSpace.norm_le_dual_bound /-
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
     Compare `continuous_linear_map.op_norm_le_bound`. -/
 theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
@@ -144,21 +157,28 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
       _ ≤ M * ‖f‖ := (hM f)
       _ = M := by rw [hf₁, mul_one]
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
+-/
 
+#print NormedSpace.eq_zero_of_forall_dual_eq_zero /-
 theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 :=
   norm_le_zero_iff.mp (norm_le_dual_bound 𝕜 x le_rfl fun f => by simp [h f])
 #align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
+-/
 
+#print NormedSpace.eq_zero_iff_forall_dual_eq_zero /-
 theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 :=
   ⟨fun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero 𝕜 h⟩
 #align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
+-/
 
+#print NormedSpace.eq_iff_forall_dual_eq /-
 /-- See also `geometric_hahn_banach_point_point`. -/
 theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g y :=
   by
   rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero 𝕜 (x - y)]
   simp [sub_eq_zero]
 #align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
+-/
 
 #print NormedSpace.inclusionInDoubleDualLi /-
 /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
@@ -195,15 +215,19 @@ variable (𝕜 : Type _) [NontriviallyNormedField 𝕜]
 
 variable {E : Type _} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
+#print NormedSpace.mem_polar_iff /-
 theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 :=
   Iff.rfl
 #align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
+-/
 
+#print NormedSpace.polar_univ /-
 @[simp]
 theorem polar_univ : polar 𝕜 (univ : Set E) = {(0 : dual 𝕜 E)} :=
   (dualPairing 𝕜 E).flip.polar_univ
     (LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft 𝕜 E))
 #align normed_space.polar_univ NormedSpace.polar_univ
+-/
 
 #print NormedSpace.isClosed_polar /-
 theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) :=
@@ -228,6 +252,7 @@ theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s :=
 
 variable {𝕜}
 
+#print NormedSpace.smul_mem_polar /-
 /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
 small scalar multiple of `x'` is in `polar 𝕜 s`. -/
 theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
@@ -245,7 +270,9 @@ theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z
     simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
   rwa [cancel] at le 
 #align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
+-/
 
+#print NormedSpace.polar_ball_subset_closedBall_div /-
 theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
     polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) :=
   by
@@ -258,9 +285,11 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
     ‖x' x‖ ≤ 1 := hx' _ h₂
     _ ≤ ‖c‖ / r * ‖x‖ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
 #align normed_space.polar_ball_subset_closed_ball_div NormedSpace.polar_ball_subset_closedBall_div
+-/
 
 variable (𝕜)
 
+#print NormedSpace.closedBall_inv_subset_polar_closedBall /-
 theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     closedBall (0 : Dual 𝕜 E) r⁻¹ ⊆ polar 𝕜 (closedBall (0 : E) r) := fun x' hx' x hx =>
   calc
@@ -271,7 +300,9 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     _ = r / r := (inv_mul_eq_div _ _)
     _ ≤ 1 := div_self_le_one r
 #align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
+-/
 
+#print NormedSpace.polar_closedBall /-
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
 inverse radius. -/
 theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
@@ -283,7 +314,9 @@ theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E]
   refine' ContinuousLinearMap.op_norm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z hz => _
   simpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.to_linear_map h z
 #align normed_space.polar_closed_ball NormedSpace.polar_closedBall
+-/
 
+#print NormedSpace.bounded_polar_of_mem_nhds_zero /-
 /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
 of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
 theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) : Bounded (polar 𝕜 s) :=
@@ -295,6 +328,7 @@ theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E))
       (((dual_pairing 𝕜 E).flip.polar_antitone r_ball).trans <|
         polar_ball_subset_closed_ball_div ha r_pos)
 #align normed_space.bounded_polar_of_mem_nhds_zero NormedSpace.bounded_polar_of_mem_nhds_zero
+-/
 
 end PolarSets
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -143,7 +143,6 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
       _ = ‖f x‖ := by rw [hfx]
       _ ≤ M * ‖f‖ := (hM f)
       _ = M := by rw [hf₁, mul_one]
-      
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 
 theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 :=
@@ -258,7 +257,6 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
   calc
     ‖x' x‖ ≤ 1 := hx' _ h₂
     _ ≤ ‖c‖ / r * ‖x‖ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
-    
 #align normed_space.polar_ball_subset_closed_ball_div NormedSpace.polar_ball_subset_closedBall_div
 
 variable (𝕜)
@@ -272,7 +270,6 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
         (dist_nonneg.trans hx'))
     _ = r / r := (inv_mul_eq_div _ _)
     _ ≤ 1 := div_self_le_one r
-    
 #align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
 
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -58,13 +58,13 @@ variable (E : Type _) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 variable (F : Type _) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 
-/- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler normed_space[normed_space] 𝕜 -/
+/- ./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler normed_space[normed_space] 𝕜 -/
 #print NormedSpace.Dual /-
 /-- The topological dual of a seminormed space `E`. -/
 def Dual :=
   E →L[𝕜] 𝕜
 deriving Inhabited, SeminormedAddCommGroup,
-  «./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler normed_space[normed_space] 𝕜»
+  «./././Mathport/Syntax/Translate/Command.lean:43:9: unsupported derive handler normed_space[normed_space] 𝕜»
 #align normed_space.dual NormedSpace.Dual
 -/
 
@@ -135,15 +135,15 @@ variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [Norm
 theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
     ‖x‖ ≤ M := by
   classical
-    by_cases h : x = 0
-    · simp only [h, hMp, norm_zero]
-    · obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h
-      calc
-        ‖x‖ = ‖(‖x‖ : 𝕜)‖ := is_R_or_C.norm_coe_norm.symm
-        _ = ‖f x‖ := by rw [hfx]
-        _ ≤ M * ‖f‖ := (hM f)
-        _ = M := by rw [hf₁, mul_one]
-        
+  by_cases h : x = 0
+  · simp only [h, hMp, norm_zero]
+  · obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h
+    calc
+      ‖x‖ = ‖(‖x‖ : 𝕜)‖ := is_R_or_C.norm_coe_norm.symm
+      _ = ‖f x‖ := by rw [hfx]
+      _ ≤ M * ‖f‖ := (hM f)
+      _ = M := by rw [hf₁, mul_one]
+      
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 
 theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -62,7 +62,8 @@ variable (F : Type _) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 #print NormedSpace.Dual /-
 /-- The topological dual of a seminormed space `E`. -/
 def Dual :=
-  E →L[𝕜] 𝕜 deriving Inhabited, SeminormedAddCommGroup,
+  E →L[𝕜] 𝕜
+deriving Inhabited, SeminormedAddCommGroup,
   «./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler normed_space[normed_space] 𝕜»
 #align normed_space.dual NormedSpace.Dual
 -/
@@ -243,14 +244,14 @@ theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z
     apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
   have cancel : ‖c⁻¹‖ * ‖c‖ = 1 := by
     simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
-  rwa [cancel] at le
+  rwa [cancel] at le 
 #align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
 
 theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
     polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) :=
   by
   intro x' hx'
-  rw [mem_polar_iff] at hx'
+  rw [mem_polar_iff] at hx' 
   simp only [polar, mem_set_of_eq, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
   have hcr : 0 < ‖c‖ / r := div_pos (zero_lt_one.trans hc) hr
   refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x h₁ h₂ => _
@@ -291,7 +292,7 @@ of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
 theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) : Bounded (polar 𝕜 s) :=
   by
   obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := NormedField.exists_one_lt_norm 𝕜
-  obtain ⟨r, r_pos, r_ball⟩ : ∃ (r : ℝ)(hr : 0 < r), ball 0 r ⊆ s := Metric.mem_nhds_iff.1 s_nhd
+  obtain ⟨r, r_pos, r_ball⟩ : ∃ (r : ℝ) (hr : 0 < r), ball 0 r ⊆ s := Metric.mem_nhds_iff.1 s_nhd
   exact
     bounded_closed_ball.mono
       (((dual_pairing 𝕜 E).flip.polar_antitone r_ball).trans <|

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -44,7 +44,7 @@ dual
 
 noncomputable section
 
-open Classical Topology
+open scoped Classical Topology
 
 universe u v
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -87,32 +87,20 @@ def inclusionInDoubleDual : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
 #align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
 -/
 
-/- warning: normed_space.dual_def -> NormedSpace.dual_def is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.dual_def NormedSpace.dual_defₓ'. -/
 @[simp]
 theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f = f x :=
   rfl
 #align normed_space.dual_def NormedSpace.dual_def
 
-/- warning: normed_space.inclusion_in_double_dual_norm_eq -> NormedSpace.inclusionInDoubleDual_norm_eq is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eqₓ'. -/
 theorem inclusionInDoubleDual_norm_eq :
     ‖inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ :=
   ContinuousLinearMap.op_norm_flip _
 #align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
 
-/- warning: normed_space.inclusion_in_double_dual_norm_le -> NormedSpace.inclusionInDoubleDual_norm_le is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_leₓ'. -/
 theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by
   rw [inclusion_in_double_dual_norm_eq]; exact ContinuousLinearMap.norm_id_le
 #align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
 
-/- warning: normed_space.double_dual_bound -> NormedSpace.double_dual_bound is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.double_dual_bound NormedSpace.double_dual_boundₓ'. -/
 theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by
   simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x
 #align normed_space.double_dual_bound NormedSpace.double_dual_bound
@@ -124,20 +112,11 @@ def dualPairing : Dual 𝕜 E →ₗ[𝕜] E →ₗ[𝕜] 𝕜 :=
 #align normed_space.dual_pairing NormedSpace.dualPairing
 -/
 
-/- warning: normed_space.dual_pairing_apply -> NormedSpace.dualPairing_apply is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.dual_pairing_apply NormedSpace.dualPairing_applyₓ'. -/
 @[simp]
 theorem dualPairing_apply {v : Dual 𝕜 E} {x : E} : dualPairing 𝕜 E v x = v x :=
   rfl
 #align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
 
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-Case conversion may be inaccurate. Consider using '#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeftₓ'. -/
 theorem dualPairing_separatingLeft : (dualPairing 𝕜 E).SeparatingLeft :=
   by
   rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
@@ -150,9 +129,6 @@ section BidualIsometry
 
 variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
-/- warning: normed_space.norm_le_dual_bound -> NormedSpace.norm_le_dual_bound is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_boundₓ'. -/
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
     Compare `continuous_linear_map.op_norm_le_bound`. -/
 theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
@@ -169,23 +145,14 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
         
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 
-/- warning: normed_space.eq_zero_of_forall_dual_eq_zero -> NormedSpace.eq_zero_of_forall_dual_eq_zero is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zeroₓ'. -/
 theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 :=
   norm_le_zero_iff.mp (norm_le_dual_bound 𝕜 x le_rfl fun f => by simp [h f])
 #align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
 
-/- warning: normed_space.eq_zero_iff_forall_dual_eq_zero -> NormedSpace.eq_zero_iff_forall_dual_eq_zero is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zeroₓ'. -/
 theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 :=
   ⟨fun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero 𝕜 h⟩
 #align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
 
-/- warning: normed_space.eq_iff_forall_dual_eq -> NormedSpace.eq_iff_forall_dual_eq is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eqₓ'. -/
 /-- See also `geometric_hahn_banach_point_point`. -/
 theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g y :=
   by
@@ -228,19 +195,10 @@ variable (𝕜 : Type _) [NontriviallyNormedField 𝕜]
 
 variable {E : Type _} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
-/- warning: normed_space.mem_polar_iff -> NormedSpace.mem_polar_iff is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.mem_polar_iff NormedSpace.mem_polar_iffₓ'. -/
 theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 :=
   Iff.rfl
 #align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
 
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 @[simp]
 theorem polar_univ : polar 𝕜 (univ : Set E) = {(0 : dual 𝕜 E)} :=
   (dualPairing 𝕜 E).flip.polar_univ
@@ -270,9 +228,6 @@ theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s :=
 
 variable {𝕜}
 
-/- warning: normed_space.smul_mem_polar -> NormedSpace.smul_mem_polar is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align normed_space.smul_mem_polar NormedSpace.smul_mem_polarₓ'. -/
 /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
 small scalar multiple of `x'` is in `polar 𝕜 s`. -/
 theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
@@ -291,12 +246,6 @@ theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z
   rwa [cancel] at le
 #align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
 
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 theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
     polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) :=
   by
@@ -313,12 +262,6 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
 
 variable (𝕜)
 
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-Case conversion may be inaccurate. Consider using '#align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBallₓ'. -/
 theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     closedBall (0 : Dual 𝕜 E) r⁻¹ ⊆ polar 𝕜 (closedBall (0 : E) r) := fun x' hx' x hx =>
   calc
@@ -331,12 +274,6 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     
 #align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
 
-/- warning: normed_space.polar_closed_ball -> NormedSpace.polar_closedBall is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align normed_space.polar_closed_ball NormedSpace.polar_closedBallₓ'. -/
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
 inverse radius. -/
 theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
@@ -349,12 +286,6 @@ theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E]
   simpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.to_linear_map h z
 #align normed_space.polar_closed_ball NormedSpace.polar_closedBall
 
-/- warning: normed_space.bounded_polar_of_mem_nhds_zero -> NormedSpace.bounded_polar_of_mem_nhds_zero is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align normed_space.bounded_polar_of_mem_nhds_zero NormedSpace.bounded_polar_of_mem_nhds_zeroₓ'. -/
 /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
 of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
 theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) : Bounded (polar 𝕜 s) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -106,10 +106,8 @@ theorem inclusionInDoubleDual_norm_eq :
 /- warning: normed_space.inclusion_in_double_dual_norm_le -> NormedSpace.inclusionInDoubleDual_norm_le is a dubious translation:
 <too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_leₓ'. -/
-theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 :=
-  by
-  rw [inclusion_in_double_dual_norm_eq]
-  exact ContinuousLinearMap.norm_id_le
+theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by
+  rw [inclusion_in_double_dual_norm_eq]; exact ContinuousLinearMap.norm_id_le
 #align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
 
 /- warning: normed_space.double_dual_bound -> NormedSpace.double_dual_bound is a dubious translation:
@@ -279,7 +277,7 @@ Case conversion may be inaccurate. Consider using '#align normed_space.smul_mem_
 small scalar multiple of `x'` is in `polar 𝕜 s`. -/
 theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
     c⁻¹ • x' ∈ polar 𝕜 s := by
-  by_cases c_zero : c = 0
+  by_cases c_zero : c = 0;
   · simp only [c_zero, inv_zero, zero_smul]
     exact (dual_pairing 𝕜 E).flip.zero_mem_polar _
   have eq : ∀ z, ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖ := fun z => norm_smul c⁻¹ _

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 
 ! This file was ported from Lean 3 source module analysis.normed_space.dual
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Analysis.LocallyConvex.Polar
 /-!
 # The topological dual of a normed space
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 In this file we define the topological dual `normed_space.dual` of a normed space, and the
 continuous linear map `normed_space.inclusion_in_double_dual` from a normed space into its double
 dual.
@@ -85,10 +88,7 @@ def inclusionInDoubleDual : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
 -/
 
 /- warning: normed_space.dual_def -> NormedSpace.dual_def is a dubious translation:
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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 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(a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) E 𝕜 (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_2))) (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 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(NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))) 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+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.dual_def NormedSpace.dual_defₓ'. -/
 @[simp]
 theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f = f x :=
@@ -96,10 +96,7 @@ theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f =
 #align normed_space.dual_def NormedSpace.dual_def
 
 /- warning: normed_space.inclusion_in_double_dual_norm_eq -> NormedSpace.inclusionInDoubleDual_norm_eq is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eqₓ'. -/
 theorem inclusionInDoubleDual_norm_eq :
     ‖inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ :=
@@ -107,10 +104,7 @@ theorem inclusionInDoubleDual_norm_eq :
 #align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_leₓ'. -/
 theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 :=
   by
@@ -119,10 +113,7 @@ theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1
 #align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
 
 /- warning: normed_space.double_dual_bound -> NormedSpace.double_dual_bound is a dubious translation:
-lean 3 declaration is
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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E 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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 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_inst_2 _inst_3))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 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(SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_2))) (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))))) (NormedSpace.inclusionInDoubleDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) x)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_2) x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.double_dual_bound NormedSpace.double_dual_boundₓ'. -/
 theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by
   simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x
@@ -136,10 +127,7 @@ def dualPairing : Dual 𝕜 E →ₗ[𝕜] E →ₗ[𝕜] 𝕜 :=
 -/
 
 /- warning: normed_space.dual_pairing_apply -> NormedSpace.dualPairing_apply is a dubious translation:
-lean 3 declaration is
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(NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (LinearMap.addCommMonoid.{u1, u1, u2, u1} 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (LinearMap.module.{u1, u1, u1, u2, u1} 𝕜 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (NormedSpace.dualPairing._proof_1.{u1} 𝕜 _inst_1))) (fun (_x : LinearMap.{u1, u1, max u2 u1, max u2 u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (LinearMap.addCommMonoid.{u1, u1, u2, u1} 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (LinearMap.module.{u1, u1, u1, u2, u1} 𝕜 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (NormedSpace.dualPairing._proof_1.{u1} 𝕜 _inst_1))) => (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) -> (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (LinearMap.hasCoeToFun.{u1, u1, max u2 u1, max u2 u1} 𝕜 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 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(NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) 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+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.dual_pairing_apply NormedSpace.dualPairing_applyₓ'. -/
 @[simp]
 theorem dualPairing_apply {v : Dual 𝕜 E} {x : E} : dualPairing 𝕜 E v x = v x :=
@@ -165,10 +153,7 @@ section BidualIsometry
 variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 /- warning: normed_space.norm_le_dual_bound -> NormedSpace.norm_le_dual_bound is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_boundₓ'. -/
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
     Compare `continuous_linear_map.op_norm_le_bound`. -/
@@ -187,30 +172,21 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 
 /- warning: normed_space.eq_zero_of_forall_dual_eq_zero -> NormedSpace.eq_zero_of_forall_dual_eq_zero is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zeroₓ'. -/
 theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 :=
   norm_le_zero_iff.mp (norm_le_dual_bound 𝕜 x le_rfl fun f => by simp [h f])
 #align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
 
 /- warning: normed_space.eq_zero_iff_forall_dual_eq_zero -> NormedSpace.eq_zero_iff_forall_dual_eq_zero is a dubious translation:
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 Case conversion may be inaccurate. Consider using '#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zeroₓ'. -/
 theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 :=
   ⟨fun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero 𝕜 h⟩
 #align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
 
 /- warning: normed_space.eq_iff_forall_dual_eq -> NormedSpace.eq_iff_forall_dual_eq is a dubious translation:
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(NormedSpace.instContinuousLinearMapClassDualToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToNormedRingToModuleToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))) g y))
+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eqₓ'. -/
 /-- See also `geometric_hahn_banach_point_point`. -/
 theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g y :=
@@ -255,10 +231,7 @@ variable (𝕜 : Type _) [NontriviallyNormedField 𝕜]
 variable {E : Type _} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 /- warning: normed_space.mem_polar_iff -> NormedSpace.mem_polar_iff is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.mem_polar_iff NormedSpace.mem_polar_iffₓ'. -/
 theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 :=
   Iff.rfl
@@ -300,10 +273,7 @@ theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s :=
 variable {𝕜}
 
 /- warning: normed_space.smul_mem_polar -> NormedSpace.smul_mem_polar is a dubious translation:
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u2} 𝕜 _inst_1 E _inst_2 _inst_3))))))) (Module.toMulActionWithZero.{u1, max u1 u2} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u1, max u1 u2} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))))))) (Inv.inv.{u1} 𝕜 (Field.toInv.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) c) x') (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 s))
+<too large>
 Case conversion may be inaccurate. Consider using '#align normed_space.smul_mem_polar NormedSpace.smul_mem_polarₓ'. -/
 /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
 small scalar multiple of `x'` is in `polar 𝕜 s`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ef95945cd48c932c9e034872bd25c3c220d9c946

@@ -56,11 +56,13 @@ variable (E : Type _) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 variable (F : Type _) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 
 /- ./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler normed_space[normed_space] 𝕜 -/
+#print NormedSpace.Dual /-
 /-- The topological dual of a seminormed space `E`. -/
 def Dual :=
   E →L[𝕜] 𝕜 deriving Inhabited, SeminormedAddCommGroup,
   «./././Mathport/Syntax/Translate/Command.lean:42:9: unsupported derive handler normed_space[normed_space] 𝕜»
 #align normed_space.dual NormedSpace.Dual
+-/
 
 instance : ContinuousLinearMapClass (Dual 𝕜 E) 𝕜 E 𝕜 :=
   ContinuousLinearMap.continuousSemilinearMapClass
@@ -74,42 +76,82 @@ instance : NormedAddCommGroup (Dual 𝕜 F) :=
 instance [FiniteDimensional 𝕜 E] : FiniteDimensional 𝕜 (Dual 𝕜 E) :=
   ContinuousLinearMap.finiteDimensional
 
+#print NormedSpace.inclusionInDoubleDual /-
 /-- The inclusion of a normed space in its double (topological) dual, considered
    as a bounded linear map. -/
 def inclusionInDoubleDual : E →L[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
   ContinuousLinearMap.apply 𝕜 𝕜
 #align normed_space.inclusion_in_double_dual NormedSpace.inclusionInDoubleDual
+-/
 
+/- warning: normed_space.dual_def -> NormedSpace.dual_def is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] (E : Type.{u2}) [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] (x : E) (f : NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3), Eq.{succ u1} 𝕜 (coeFn.{max (succ (max u2 u1)) (succ u1), max (succ (max u2 u1)) (succ u1)} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (fun (_x : NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) => (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u1, max u2 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(NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.Dual.normedSpace.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 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+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] (E : Type.{u1}) [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2] (x : E) (f : NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3), Eq.{succ u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) => 𝕜) f) (FunLike.coe.{max (succ u2) (succ (max u2 u1)), succ (max u2 u1), succ u2} (NormedSpace.Dual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (fun (_x : NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) => (fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : 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+Case conversion may be inaccurate. Consider using '#align normed_space.dual_def NormedSpace.dual_defₓ'. -/
 @[simp]
 theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f = f x :=
   rfl
 #align normed_space.dual_def NormedSpace.dual_def
 
+/- warning: normed_space.inclusion_in_double_dual_norm_eq -> NormedSpace.inclusionInDoubleDual_norm_eq is a dubious translation:
+lean 3 declaration is
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_inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.Dual.normedSpace.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))) (ContinuousLinearMap.hasOpNorm.{u1, u1, u2, max u2 u1} 𝕜 𝕜 E (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) _inst_2 (NormedSpace.Dual.seminormedAddCommGroup.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) _inst_1 _inst_1 _inst_3 (NormedSpace.Dual.normedSpace.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.inclusionInDoubleDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Norm.norm.{max u2 u1} (ContinuousLinearMap.{u1, u1, max u2 u1, max u2 u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (ContinuousLinearMap.hasOpNorm.{u1, u1, max u2 u1, max u2 u1} 𝕜 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) _inst_1 _inst_1 (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (ContinuousLinearMap.id.{u1, max u2 u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))))
+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] (E : Type.{u1}) [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2], Eq.{1} Real (Norm.norm.{max u2 u1} (ContinuousLinearMap.{u2, u2, u1, max u1 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_2))) (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u2, max u2 u1} 𝕜 (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) (ContinuousLinearMap.hasOpNorm.{u2, u2, u1, max u2 u1} 𝕜 𝕜 E (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) _inst_2 (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) _inst_1 _inst_1 _inst_3 (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.inclusionInDoubleDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (Norm.norm.{max u2 u1} (ContinuousLinearMap.{u2, u2, max u1 u2, max u1 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u2, max u2 u1} 𝕜 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.toModule.{u2, max u2 u1} 𝕜 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))) (ContinuousLinearMap.hasOpNorm.{u2, u2, max u2 u1, max u2 u1} 𝕜 𝕜 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) _inst_1 _inst_1 (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (ContinuousLinearMap.id.{u2, max u1 u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u2, max u2 u1} 𝕜 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))))
+Case conversion may be inaccurate. Consider using '#align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eqₓ'. -/
 theorem inclusionInDoubleDual_norm_eq :
     ‖inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ :=
   ContinuousLinearMap.op_norm_flip _
 #align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
 
+/- warning: normed_space.inclusion_in_double_dual_norm_le -> NormedSpace.inclusionInDoubleDual_norm_le is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] (E : Type.{u2}) [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2], LE.le.{0} Real Real.hasLe (Norm.norm.{max u2 u1} (ContinuousLinearMap.{u1, u1, u2, max u2 u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E _inst_2))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 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+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] (E : Type.{u1}) [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2], LE.le.{0} Real Real.instLEReal (Norm.norm.{max u2 u1} (ContinuousLinearMap.{u2, u2, u1, max u1 u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E 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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) (ContinuousLinearMap.hasOpNorm.{u2, u2, u1, max u2 u1} 𝕜 𝕜 E (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) _inst_2 (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) _inst_1 _inst_1 _inst_3 (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 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+Case conversion may be inaccurate. Consider using '#align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_leₓ'. -/
 theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 :=
   by
   rw [inclusion_in_double_dual_norm_eq]
   exact ContinuousLinearMap.norm_id_le
 #align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
 
+/- warning: normed_space.double_dual_bound -> NormedSpace.double_dual_bound is a dubious translation:
+lean 3 declaration is
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(NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (UniformSpace.toTopologicalSpace.{max u2 u1} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (PseudoMetricSpace.toUniformSpace.{max u2 u1} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.Dual.seminormedAddCommGroup.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.Dual.seminormedAddCommGroup.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.Dual.normedSpace.{u1, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))) (NormedSpace.inclusionInDoubleDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) x)) (Norm.norm.{u2} E (SeminormedAddCommGroup.toHasNorm.{u2} E _inst_2) x)
+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] (E : Type.{u1}) [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2] (x : E), LE.le.{0} Real Real.instLEReal (Norm.norm.{max u2 u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) x) (SeminormedAddCommGroup.toNorm.{max u2 u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E 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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 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_inst_2 _inst_3))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, max u1 u2} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.instSeminormedAddCommGroupDual.{u2, max u2 u1} 𝕜 _inst_1 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 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(NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)))))) (NormedSpace.inclusionInDoubleDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) x)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_2) x)
+Case conversion may be inaccurate. Consider using '#align normed_space.double_dual_bound NormedSpace.double_dual_boundₓ'. -/
 theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by
   simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusion_in_double_dual_norm_le 𝕜 E) x
 #align normed_space.double_dual_bound NormedSpace.double_dual_bound
 
+#print NormedSpace.dualPairing /-
 /-- The dual pairing as a bilinear form. -/
 def dualPairing : Dual 𝕜 E →ₗ[𝕜] E →ₗ[𝕜] 𝕜 :=
   ContinuousLinearMap.coeLM 𝕜
 #align normed_space.dual_pairing NormedSpace.dualPairing
+-/
 
+/- warning: normed_space.dual_pairing_apply -> NormedSpace.dualPairing_apply is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] (E : Type.{u2}) [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] {v : NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3} {x : E}, Eq.{succ u1} 𝕜 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 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(NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (LinearMap.addCommMonoid.{u1, u1, u2, u1} 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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (LinearMap.module.{u1, u1, u1, u2, u1} 𝕜 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (NormedSpace.dualPairing._proof_1.{u1} 𝕜 _inst_1))) (fun (_x : LinearMap.{u1, u1, max u2 u1, max u2 u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (LinearMap.addCommMonoid.{u1, u1, u2, u1} 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (LinearMap.module.{u1, u1, u1, u2, u1} 𝕜 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (NormedSpace.dualPairing._proof_1.{u1} 𝕜 _inst_1))) => (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) -> (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (LinearMap.hasCoeToFun.{u1, u1, max u2 u1, max u2 u1} 𝕜 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (LinearMap.{u1, u1, u2, u1} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (LinearMap.addCommMonoid.{u1, u1, u2, u1} 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (LinearMap.module.{u1, u1, u1, u2, u1} 𝕜 𝕜 𝕜 E 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (NormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNormedAddCommGroup.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (NormedSpace.dualPairing._proof_1.{u1} 𝕜 _inst_1)) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.dualPairing.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) v) x) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (fun (_x : NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) => E -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) v x)
+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] (E : Type.{u1}) [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2] {v : NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3} {x : E}, Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : E) => 𝕜) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) => LinearMap.{u2, u2, u1, u2} 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 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(NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) E 𝕜 (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) 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(NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u2, max u2 u1} 𝕜 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (LinearMap.instModuleLinearMapAddCommMonoid.{u2, u2, u2, u1, u2} 𝕜 𝕜 𝕜 E 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))) (smulCommClass_self.{u2, u2} 𝕜 𝕜 (CommRing.toCommMonoid.{u2} 𝕜 (EuclideanDomain.toCommRing.{u2} 𝕜 (Field.toEuclideanDomain.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (MulActionWithZero.toMulAction.{u2, u2} 𝕜 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) (MonoidWithZero.toMulActionWithZero.{u2} 𝕜 (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.dualPairing.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) v) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) E 𝕜 (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_2))) (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_2))) (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) 𝕜 (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))) (NormedSpace.instContinuousLinearMapClassDualToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToNormedRingToModuleToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))) v x)
+Case conversion may be inaccurate. Consider using '#align normed_space.dual_pairing_apply NormedSpace.dualPairing_applyₓ'. -/
 @[simp]
 theorem dualPairing_apply {v : Dual 𝕜 E} {x : E} : dualPairing 𝕜 E v x = v x :=
   rfl
 #align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
 
+/- warning: normed_space.dual_pairing_separating_left -> NormedSpace.dualPairing_separatingLeft is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] (E : Type.{u2}) [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2], LinearMap.SeparatingLeft.{u1, u1, u1, max u2 u1, u2} 𝕜 𝕜 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) E (Semifield.toCommSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (Semifield.toCommSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Semifield.toCommSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NormedSpace.dualPairing.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)
+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] (E : Type.{u1}) [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2], LinearMap.SeparatingLeft.{u2, u2, u2, max u2 u1, u1} 𝕜 𝕜 𝕜 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) E (Semifield.toCommSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))) (Semifield.toCommSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u2, max u2 u1} 𝕜 (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (Semifield.toCommSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)))) (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2 _inst_3) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1))))))) (NormedSpace.dualPairing.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)
+Case conversion may be inaccurate. Consider using '#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeftₓ'. -/
 theorem dualPairing_separatingLeft : (dualPairing 𝕜 E).SeparatingLeft :=
   by
   rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
@@ -122,6 +164,12 @@ section BidualIsometry
 
 variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
+/- warning: normed_space.norm_le_dual_bound -> NormedSpace.norm_le_dual_bound is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u2}) [_inst_1 : IsROrC.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] (x : E) {M : Real}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) M) -> (forall (f : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3), LE.le.{0} Real Real.hasLe (Norm.norm.{u2} 𝕜 (NormedField.toHasNorm.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (fun (_x : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) => E -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) f x)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) M (Norm.norm.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedAddCommGroup.toHasNorm.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.Dual.normedAddCommGroup.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E _inst_2 _inst_3)) f))) -> (LE.le.{0} Real Real.hasLe (Norm.norm.{u1} E (NormedAddCommGroup.toHasNorm.{u1} E _inst_2) x) M)
+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : IsROrC.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] (x : E) {M : Real}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) M) -> (forall (f : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3), LE.le.{0} Real Real.instLEReal (Norm.norm.{u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) x) (NormedField.toNorm.{u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) x) (DenselyNormedField.toNormedField.{u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) x) (IsROrC.toDenselyNormedField.{u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) x) _inst_1))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) E 𝕜 (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))))))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) 𝕜 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))))) (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))))) (RingHom.id.{u2} 𝕜 (Semiring.toNonAssocSemiring.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2))) 𝕜 (UniformSpace.toTopologicalSpace.{u2} 𝕜 (PseudoMetricSpace.toUniformSpace.{u2} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u2} 𝕜 (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)))))))))) (NormedSpace.toModule.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)))))))) (NormedField.toNormedSpace.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))) (NormedSpace.instContinuousLinearMapClassDualToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToNormedRingToModuleToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))) f x)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) M (Norm.norm.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedAddCommGroup.toNorm.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.instNormedAddCommGroupDualToSeminormedAddCommGroup.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E _inst_2 _inst_3)) f))) -> (LE.le.{0} Real Real.instLEReal (Norm.norm.{u1} E (NormedAddCommGroup.toNorm.{u1} E _inst_2) x) M)
+Case conversion may be inaccurate. Consider using '#align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_boundₓ'. -/
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
     Compare `continuous_linear_map.op_norm_le_bound`. -/
 theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
@@ -138,14 +186,32 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
         
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 
+/- warning: normed_space.eq_zero_of_forall_dual_eq_zero -> NormedSpace.eq_zero_of_forall_dual_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u2}) [_inst_1 : IsROrC.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E}, (forall (f : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3), Eq.{succ u2} 𝕜 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (fun (_x : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) => E -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) f x) (OfNat.ofNat.{u2} 𝕜 0 (OfNat.mk.{u2} 𝕜 0 (Zero.zero.{u2} 𝕜 (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)))))))))))))) -> (Eq.{succ u1} E x (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (NormedAddGroup.toAddGroup.{u1} E (NormedAddCommGroup.toNormedAddGroup.{u1} E _inst_2))))))))))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zeroₓ'. -/
 theorem eq_zero_of_forall_dual_eq_zero {x : E} (h : ∀ f : Dual 𝕜 E, f x = (0 : 𝕜)) : x = 0 :=
   norm_le_zero_iff.mp (norm_le_dual_bound 𝕜 x le_rfl fun f => by simp [h f])
 #align normed_space.eq_zero_of_forall_dual_eq_zero NormedSpace.eq_zero_of_forall_dual_eq_zero
 
+/- warning: normed_space.eq_zero_iff_forall_dual_eq_zero -> NormedSpace.eq_zero_iff_forall_dual_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u2}) [_inst_1 : IsROrC.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] (x : E), Iff (Eq.{succ u1} E x (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (NormedAddGroup.toAddGroup.{u1} E (NormedAddCommGroup.toNormedAddGroup.{u1} E _inst_2)))))))))) (forall (g : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3), Eq.{succ u2} 𝕜 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (fun (_x : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) => E -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) g x) (OfNat.ofNat.{u2} 𝕜 0 (OfNat.mk.{u2} 𝕜 0 (Zero.zero.{u2} 𝕜 (MulZeroClass.toHasZero.{u2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u2} 𝕜 (Ring.toNonAssocRing.{u2} 𝕜 (NormedRing.toRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))))))))))))
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(NormedSpace.instContinuousLinearMapClassDualToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToNormedRingToModuleToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))) g x) (OfNat.ofNat.{u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) x) 0 (Zero.toOfNat0.{u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) x) (CommMonoidWithZero.toZero.{u2} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) x) (CommGroupWithZero.toCommMonoidWithZero.{u2} 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+Case conversion may be inaccurate. Consider using '#align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zeroₓ'. -/
 theorem eq_zero_iff_forall_dual_eq_zero (x : E) : x = 0 ↔ ∀ g : Dual 𝕜 E, g x = 0 :=
   ⟨fun hx => by simp [hx], fun h => eq_zero_of_forall_dual_eq_zero 𝕜 h⟩
 #align normed_space.eq_zero_iff_forall_dual_eq_zero NormedSpace.eq_zero_iff_forall_dual_eq_zero
 
+/- warning: normed_space.eq_iff_forall_dual_eq -> NormedSpace.eq_iff_forall_dual_eq is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u2}) [_inst_1 : IsROrC.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {y : E}, Iff (Eq.{succ u1} E x y) (forall (g : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3), Eq.{succ u2} 𝕜 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (fun (_x : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) => E -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) g x) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (fun (_x : NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) => E -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) g y))
+but is expected to have type
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(DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u2} 𝕜 (NormedRing.toNonUnitalNormedRing.{u2} 𝕜 (NormedCommRing.toNormedRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)))))))) (NormedField.toNormedSpace.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1))))) (NormedSpace.instContinuousLinearMapClassDualToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToNormedRingToModuleToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_1)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))) g y))
+Case conversion may be inaccurate. Consider using '#align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eqₓ'. -/
 /-- See also `geometric_hahn_banach_point_point`. -/
 theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g y :=
   by
@@ -153,6 +219,7 @@ theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g
   simp [sub_eq_zero]
 #align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
 
+#print NormedSpace.inclusionInDoubleDualLi /-
 /-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
 def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
   { inclusionInDoubleDual 𝕜 E with
@@ -165,6 +232,7 @@ def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
       rintro c ⟨hc1, hc2⟩
       exact norm_le_dual_bound 𝕜 x hc1 hc2 }
 #align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
+-/
 
 end BidualIsometry
 
@@ -172,6 +240,7 @@ section PolarSets
 
 open Metric Set NormedSpace
 
+#print NormedSpace.polar /-
 /-- Given a subset `s` in a normed space `E` (over a field `𝕜`), the polar
 `polar 𝕜 s` is the subset of `dual 𝕜 E` consisting of those functionals which
 evaluate to something of norm at most one at all points `z ∈ s`. -/
@@ -179,21 +248,35 @@ def polar (𝕜 : Type _) [NontriviallyNormedField 𝕜] {E : Type _} [Seminorme
     [NormedSpace 𝕜 E] : Set E → Set (Dual 𝕜 E) :=
   (dualPairing 𝕜 E).flip.polar
 #align normed_space.polar NormedSpace.polar
+-/
 
 variable (𝕜 : Type _) [NontriviallyNormedField 𝕜]
 
 variable {E : Type _} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
+/- warning: normed_space.mem_polar_iff -> NormedSpace.mem_polar_iff is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align normed_space.mem_polar_iff NormedSpace.mem_polar_iffₓ'. -/
 theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 :=
   Iff.rfl
 #align normed_space.mem_polar_iff NormedSpace.mem_polar_iff
 
+/- warning: normed_space.polar_univ -> NormedSpace.polar_univ is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2], Eq.{succ (max u2 u1)} (Set.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 (Set.univ.{u2} E)) (Singleton.singleton.{max u2 u1, max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (Set.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.hasSingleton.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (OfNat.mk.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (Zero.zero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddZeroClass.toHasZero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddMonoid.toAddZeroClass.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SubNegMonoid.toAddMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddGroup.toSubNegMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddGroup.toAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toSeminormedAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))))))))))
+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2], Eq.{max (succ u2) (succ u1)} (Set.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.polar.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3 (Set.univ.{u1} E)) (Singleton.singleton.{max u2 u1, max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (Set.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.instSingletonSet.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (Zero.toOfNat0.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NegZeroClass.toZero.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubNegZeroMonoid.toNegZeroClass.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionMonoid.toSubNegZeroMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionCommMonoid.toSubtractionMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommGroup.toDivisionAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))))))))))
+Case conversion may be inaccurate. Consider using '#align normed_space.polar_univ NormedSpace.polar_univₓ'. -/
 @[simp]
 theorem polar_univ : polar 𝕜 (univ : Set E) = {(0 : dual 𝕜 E)} :=
   (dualPairing 𝕜 E).flip.polar_univ
     (LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft 𝕜 E))
 #align normed_space.polar_univ NormedSpace.polar_univ
 
+#print NormedSpace.isClosed_polar /-
 theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) :=
   by
   dsimp only [NormedSpace.polar]
@@ -201,7 +284,9 @@ theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) :=
   refine' isClosed_biInter fun z hz => _
   exact is_closed_Iic.preimage (ContinuousLinearMap.apply 𝕜 𝕜 z).Continuous.norm
 #align normed_space.is_closed_polar NormedSpace.isClosed_polar
+-/
 
+#print NormedSpace.polar_closure /-
 @[simp]
 theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s :=
   ((dualPairing 𝕜 E).flip.polar_antitone subset_closure).antisymm <|
@@ -210,9 +295,16 @@ theorem polar_closure (s : Set E) : polar 𝕜 (closure s) = polar 𝕜 s :=
         simpa [LinearMap.flip_flip] using
           (is_closed_polar _ _).Preimage (inclusion_in_double_dual 𝕜 E).Continuous
 #align normed_space.polar_closure NormedSpace.polar_closure
+-/
 
 variable {𝕜}
 
+/- warning: normed_space.smul_mem_polar -> NormedSpace.smul_mem_polar is a dubious translation:
+lean 3 declaration is
+  forall {𝕜 : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] {s : Set.{u2} E} {x' : NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3} {c : 𝕜}, (forall (z : E), (Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) z s) -> (LE.le.{0} Real Real.hasLe (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (fun (_x : NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) => E -> 𝕜) (NormedSpace.Dual.hasCoeToFun.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) x' z)) (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)) c))) -> (Membership.Mem.{max u2 u1, max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (Set.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.hasMem.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (SMul.smul.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SMulZeroClass.toHasSmul.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddZeroClass.toHasZero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddMonoid.toAddZeroClass.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommMonoid.toAddMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))))) (AddZeroClass.toHasZero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddMonoid.toAddZeroClass.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommMonoid.toAddMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (AddZeroClass.toHasZero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddMonoid.toAddZeroClass.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommMonoid.toAddMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))))) (Module.toMulActionWithZero.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u1, max u2 u1} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.normedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))))) (Inv.inv.{u1} 𝕜 (DivInvMonoid.toHasInv.{u1} 𝕜 (DivisionRing.toDivInvMonoid.{u1} 𝕜 (NormedDivisionRing.toDivisionRing.{u1} 𝕜 (NormedField.toNormedDivisionRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) c) x') (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 s))
+but is expected to have type
+  forall {𝕜 : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] {s : Set.{u2} E} {x' : NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3} {c : 𝕜}, (forall (z : E), (Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) z s) -> (LE.le.{0} Real Real.instLEReal (Norm.norm.{u1} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) z) (NormedField.toNorm.{u1} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) z) (NontriviallyNormedField.toNormedField.{u1} ((fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) z) _inst_1)) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.NormedSpace.Dual._hyg.279 : E) => 𝕜) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u2, u1} 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(NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E _inst_2))) (AddCommGroup.toAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) 𝕜 (UniformSpace.toTopologicalSpace.{u1} 𝕜 (PseudoMetricSpace.toUniformSpace.{u1} 𝕜 (SeminormedRing.toPseudoMetricSpace.{u1} 𝕜 (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} 𝕜 (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} 𝕜 (NonAssocRing.toNonUnitalNonAssocRing.{u1} 𝕜 (Ring.toNonAssocRing.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))))) (NormedSpace.toModule.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2 _inst_3) (NormedSpace.toModule.{u1, u1} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NormedField.toNormedSpace.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) (NormedSpace.instContinuousLinearMapClassDualToSemiringToDivisionSemiringToSemifieldToFieldToNormedFieldToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToAddCommMonoidToAddCommGroupToTopologicalSpaceToUniformSpaceToPseudoMetricSpaceToSeminormedRingToSeminormedCommRingToNormedCommRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToNormedRingToModuleToModuleToSeminormedAddCommGroupToNonUnitalSeminormedRingToNonUnitalNormedRingToNormedSpace.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) x' z)) (Norm.norm.{u1} 𝕜 (NormedField.toNorm.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)) c))) -> (Membership.mem.{max u1 u2, max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (Set.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.instMembershipSet.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (HSMul.hSMul.{u1, max u1 u2, max u1 u2} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 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(MulActionWithZero.toSMulWithZero.{u1, max u1 u2} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)))))) (NegZeroClass.toZero.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SubNegZeroMonoid.toNegZeroClass.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionMonoid.toSubNegZeroMonoid.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionCommMonoid.toSubtractionMonoid.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommGroup.toDivisionAddCommMonoid.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))))))) (Module.toMulActionWithZero.{u1, max u1 u2} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))))) (AddCommGroup.toAddCommMonoid.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))) (NormedSpace.toModule.{u1, max u1 u2} 𝕜 (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) (NormedSpace.instSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instNormedSpaceDualToNormedFieldInstSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3))))))) (Inv.inv.{u1} 𝕜 (Field.toInv.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1))) c) x') (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 s))
+Case conversion may be inaccurate. Consider using '#align normed_space.smul_mem_polar NormedSpace.smul_mem_polarₓ'. -/
 /-- If `x'` is a dual element such that the norms `‖x' z‖` are bounded for `z ∈ s`, then a
 small scalar multiple of `x'` is in `polar 𝕜 s`. -/
 theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z ∈ s → ‖x' z‖ ≤ ‖c‖) :
@@ -231,6 +323,12 @@ theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z
   rwa [cancel] at le
 #align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
 
+/- warning: normed_space.polar_ball_subset_closed_ball_div -> NormedSpace.polar_ball_subset_closedBall_div is a dubious translation:
+lean 3 declaration is
+  forall {𝕜 : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] {c : 𝕜}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)) c)) -> (forall {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (HasSubset.Subset.{max u2 u1} (Set.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.hasSubset.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 (Metric.ball.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E _inst_2) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (SeminormedAddGroup.toAddGroup.{u2} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} E _inst_2))))))))) r)) (Metric.closedBall.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (OfNat.mk.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (Zero.zero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddZeroClass.toHasZero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddMonoid.toAddZeroClass.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SubNegMonoid.toAddMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddGroup.toSubNegMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddGroup.toAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toSeminormedAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))))))))) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1)) c) r))))
+but is expected to have type
+  forall {𝕜 : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2] {c : 𝕜}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (Norm.norm.{u2} 𝕜 (NormedField.toNorm.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)) c)) -> (forall {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (HasSubset.Subset.{max u2 u1} (Set.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.instHasSubsetSet.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.polar.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3 (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_2) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)))))))) r)) (Metric.closedBall.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (Zero.toOfNat0.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NegZeroClass.toZero.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubNegZeroMonoid.toNegZeroClass.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionMonoid.toSubNegZeroMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionCommMonoid.toSubtractionMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommGroup.toDivisionAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))))))))) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (Norm.norm.{u2} 𝕜 (NormedField.toNorm.{u2} 𝕜 (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1)) c) r))))
+Case conversion may be inaccurate. Consider using '#align normed_space.polar_ball_subset_closed_ball_div NormedSpace.polar_ball_subset_closedBall_divₓ'. -/
 theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ} (hr : 0 < r) :
     polar 𝕜 (ball (0 : E) r) ⊆ closedBall (0 : Dual 𝕜 E) (‖c‖ / r) :=
   by
@@ -247,6 +345,12 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
 
 variable (𝕜)
 
+/- warning: normed_space.closed_ball_inv_subset_polar_closed_ball -> NormedSpace.closedBall_inv_subset_polar_closedBall is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] {r : Real}, HasSubset.Subset.{max u2 u1} (Set.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.hasSubset.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (Metric.closedBall.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (OfNat.mk.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (Zero.zero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddZeroClass.toHasZero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddMonoid.toAddZeroClass.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SubNegMonoid.toAddMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (AddGroup.toSubNegMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddGroup.toAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toSeminormedAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)))))))))) (Inv.inv.{0} Real Real.hasInv r)) (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 (Metric.closedBall.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E _inst_2) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (SeminormedAddGroup.toAddGroup.{u2} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} E _inst_2))))))))) r))
+but is expected to have type
+  forall (𝕜 : Type.{u2}) [_inst_1 : NontriviallyNormedField.{u2} 𝕜] {E : Type.{u1}} [_inst_2 : SeminormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} 𝕜 E (NontriviallyNormedField.toNormedField.{u2} 𝕜 _inst_1) _inst_2] {r : Real}, HasSubset.Subset.{max u2 u1} (Set.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (Set.instHasSubsetSet.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (Metric.closedBall.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) 0 (Zero.toOfNat0.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NegZeroClass.toZero.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubNegZeroMonoid.toNegZeroClass.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionMonoid.toSubNegZeroMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SubtractionCommMonoid.toSubtractionMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (AddCommGroup.toDivisionAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3))))))))) (Inv.inv.{0} Real Real.instInvReal r)) (NormedSpace.polar.{u2, u1} 𝕜 _inst_1 E _inst_2 _inst_3 (Metric.closedBall.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_2) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)))))))) r))
+Case conversion may be inaccurate. Consider using '#align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBallₓ'. -/
 theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     closedBall (0 : Dual 𝕜 E) r⁻¹ ⊆ polar 𝕜 (closedBall (0 : E) r) := fun x' hx' x hx =>
   calc
@@ -259,6 +363,12 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     
 #align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
 
+/- warning: normed_space.polar_closed_ball -> NormedSpace.polar_closedBall is a dubious translation:
+lean 3 declaration is
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_4 : IsROrC.{u1} 𝕜] [_inst_5 : NormedAddCommGroup.{u2} E] [_inst_6 : NormedSpace.{u1, u2} 𝕜 E (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)] {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (Eq.{succ (max u2 u1)} (Set.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) (NormedSpace.polar.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6 (Metric.closedBall.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5)) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (NormedAddGroup.toAddGroup.{u2} E (NormedAddCommGroup.toNormedAddGroup.{u2} E _inst_5))))))))) r)) (Metric.closedBall.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) 0 (OfNat.mk.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) 0 (Zero.zero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (AddZeroClass.toHasZero.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (AddMonoid.toAddZeroClass.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (SubNegMonoid.toAddMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (AddGroup.toSubNegMonoid.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (NormedAddGroup.toAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (NormedAddCommGroup.toNormedAddGroup.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_5) _inst_6) (NormedSpace.Dual.normedAddCommGroup.{u1, u2} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_4)) E _inst_5 _inst_6)))))))))) (Inv.inv.{0} Real Real.hasInv r)))
+but is expected to have type
+  forall {𝕜 : Type.{u2}} {E : Type.{u1}} [_inst_4 : IsROrC.{u2} 𝕜] [_inst_5 : NormedAddCommGroup.{u1} E] [_inst_6 : NormedSpace.{u2, u1} 𝕜 E (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)] {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (Eq.{max (succ u2) (succ u1)} (Set.{max u1 u2} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) (NormedSpace.polar.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6 (Metric.closedBall.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5)) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_5)))))))) r)) (Metric.closedBall.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (NormedSpace.instSeminormedAddCommGroupDual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6)) (OfNat.ofNat.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) 0 (Zero.toOfNat0.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (NegZeroClass.toZero.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (SubNegZeroMonoid.toNegZeroClass.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (SubtractionMonoid.toSubNegZeroMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (SubtractionCommMonoid.toSubtractionMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (AddCommGroup.toDivisionAddCommMonoid.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (NormedAddCommGroup.toAddCommGroup.{max u2 u1} (NormedSpace.Dual.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_5) _inst_6) (NormedSpace.instNormedAddCommGroupDualToSeminormedAddCommGroup.{u2, u1} 𝕜 (DenselyNormedField.toNontriviallyNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_4)) E _inst_5 _inst_6))))))))) (Inv.inv.{0} Real Real.instInvReal r)))
+Case conversion may be inaccurate. Consider using '#align normed_space.polar_closed_ball NormedSpace.polar_closedBallₓ'. -/
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
 inverse radius. -/
 theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
@@ -271,6 +381,12 @@ theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E]
   simpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.to_linear_map h z
 #align normed_space.polar_closed_ball NormedSpace.polar_closedBall
 
+/- warning: normed_space.bounded_polar_of_mem_nhds_zero -> NormedSpace.bounded_polar_of_mem_nhds_zero is a dubious translation:
+lean 3 declaration is
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] {s : Set.{u2} E}, (Membership.Mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (Filter.hasMem.{u2} E) s (nhds.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E _inst_2))) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (SeminormedAddGroup.toAddGroup.{u2} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u2} E _inst_2))))))))))) -> (Metric.Bounded.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u2 u1} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.Dual.seminormedAddCommGroup.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 s))
+but is expected to have type
+  forall (𝕜 : Type.{u1}) [_inst_1 : NontriviallyNormedField.{u1} 𝕜] {E : Type.{u2}} [_inst_2 : SeminormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} 𝕜 E (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_1) _inst_2] {s : Set.{u2} E}, (Membership.mem.{u2, u2} (Set.{u2} E) (Filter.{u2} E) (instMembershipSetFilter.{u2} E) s (nhds.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E _inst_2))) (OfNat.ofNat.{u2} E 0 (Zero.toOfNat0.{u2} E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E (SeminormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)))))))))) -> (Metric.Bounded.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (SeminormedAddCommGroup.toPseudoMetricSpace.{max u1 u2} (NormedSpace.Dual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3) (NormedSpace.instSeminormedAddCommGroupDual.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3)) (NormedSpace.polar.{u1, u2} 𝕜 _inst_1 E _inst_2 _inst_3 s))
+Case conversion may be inaccurate. Consider using '#align normed_space.bounded_polar_of_mem_nhds_zero NormedSpace.bounded_polar_of_mem_nhds_zeroₓ'. -/
 /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
 of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
 theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) : Bounded (polar 𝕜 s) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -161,7 +161,7 @@ def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
       apply le_antisymm
       · exact double_dual_bound 𝕜 E x
       rw [ContinuousLinearMap.norm_def]
-      refine' le_cinfₛ ContinuousLinearMap.bounds_nonempty _
+      refine' le_csInf ContinuousLinearMap.bounds_nonempty _
       rintro c ⟨hc1, hc2⟩
       exact norm_le_dual_bound 𝕜 x hc1 hc2 }
 #align normed_space.inclusion_in_double_dual_li NormedSpace.inclusionInDoubleDualLi
@@ -197,8 +197,8 @@ theorem polar_univ : polar 𝕜 (univ : Set E) = {(0 : dual 𝕜 E)} :=
 theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) :=
   by
   dsimp only [NormedSpace.polar]
-  simp only [LinearMap.polar_eq_interᵢ, LinearMap.flip_apply]
-  refine' isClosed_binterᵢ fun z hz => _
+  simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
+  refine' isClosed_biInter fun z hz => _
   exact is_closed_Iic.preimage (ContinuousLinearMap.apply 𝕜 𝕜 z).Continuous.norm
 #align normed_space.is_closed_polar NormedSpace.isClosed_polar
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/3cacc945118c8c637d89950af01da78307f59325

@@ -102,7 +102,7 @@ theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ 
 
 /-- The dual pairing as a bilinear form. -/
 def dualPairing : Dual 𝕜 E →ₗ[𝕜] E →ₗ[𝕜] 𝕜 :=
-  ContinuousLinearMap.coeLm 𝕜
+  ContinuousLinearMap.coeLM 𝕜
 #align normed_space.dual_pairing NormedSpace.dualPairing
 
 @[simp]

mathlib commit https://github.com/leanprover-community/mathlib/commit/dd6388c44e6f6b4547070b887c5905d5cfe6c9f8

@@ -110,11 +110,11 @@ theorem dualPairing_apply {v : Dual 𝕜 E} {x : E} : dualPairing 𝕜 E v x = v
   rfl
 #align normed_space.dual_pairing_apply NormedSpace.dualPairing_apply
 
-theorem dualPairingSeparatingLeft : (dualPairing 𝕜 E).SeparatingLeft :=
+theorem dualPairing_separatingLeft : (dualPairing 𝕜 E).SeparatingLeft :=
   by
   rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
   exact ContinuousLinearMap.coe_injective
-#align normed_space.dual_pairing_separating_left NormedSpace.dualPairingSeparatingLeft
+#align normed_space.dual_pairing_separating_left NormedSpace.dualPairing_separatingLeft
 
 end General
 
@@ -191,7 +191,7 @@ theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ 
 @[simp]
 theorem polar_univ : polar 𝕜 (univ : Set E) = {(0 : dual 𝕜 E)} :=
   (dualPairing 𝕜 E).flip.polar_univ
-    (LinearMap.flip_separatingRight.mpr (dualPairingSeparatingLeft 𝕜 E))
+    (LinearMap.flip_separatingRight.mpr (dualPairing_separatingLeft 𝕜 E))
 #align normed_space.polar_univ NormedSpace.polar_univ
 
 theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba

@@ -250,7 +250,7 @@ variable (𝕜)
 theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     closedBall (0 : Dual 𝕜 E) r⁻¹ ⊆ polar 𝕜 (closedBall (0 : E) r) := fun x' hx' x hx =>
   calc
-    ‖x' x‖ ≤ ‖x'‖ * ‖x‖ := x'.le_op_norm x
+    ‖x' x‖ ≤ ‖x'‖ * ‖x‖ := x'.le_opNorm x
     _ ≤ r⁻¹ * r :=
       (mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
         (dist_nonneg.trans hx'))

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -133,7 +133,7 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
       calc
         ‖x‖ = ‖(‖x‖ : 𝕜)‖ := is_R_or_C.norm_coe_norm.symm
         _ = ‖f x‖ := by rw [hfx]
-        _ ≤ M * ‖f‖ := hM f
+        _ ≤ M * ‖f‖ := (hM f)
         _ = M := by rw [hf₁, mul_one]
         
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
@@ -252,9 +252,9 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
   calc
     ‖x' x‖ ≤ ‖x'‖ * ‖x‖ := x'.le_op_norm x
     _ ≤ r⁻¹ * r :=
-      mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
-        (dist_nonneg.trans hx')
-    _ = r / r := inv_mul_eq_div _ _
+      (mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
+        (dist_nonneg.trans hx'))
+    _ = r / r := (inv_mul_eq_div _ _)
     _ ≤ 1 := div_self_le_one r
     
 #align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -120,7 +120,7 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
       calc
         ‖x‖ = ‖(‖x‖ : 𝕜)‖ := RCLike.norm_coe_norm.symm
         _ = ‖f x‖ := by rw [hfx]
-        _ ≤ M * ‖f‖ := (hM f)
+        _ ≤ M * ‖f‖ := hM f
         _ = M := by rw [hf₁, mul_one]
 #align normed_space.norm_le_dual_bound NormedSpace.norm_le_dual_bound
 
@@ -234,7 +234,7 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     _ ≤ r⁻¹ * r :=
       (mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
         (dist_nonneg.trans hx'))
-    _ = r / r := (inv_mul_eq_div _ _)
+    _ = r / r := inv_mul_eq_div _ _
     _ ≤ 1 := div_self_le_one r
 #align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
 

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -138,7 +138,7 @@ theorem eq_iff_forall_dual_eq {x y : E} : x = y ↔ ∀ g : Dual 𝕜 E, g x = g
   simp [sub_eq_zero]
 #align normed_space.eq_iff_forall_dual_eq NormedSpace.eq_iff_forall_dual_eq
 
-/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/
+/-- The inclusion of a normed space in its double dual is an isometry onto its image. -/
 def inclusionInDoubleDualLi : E →ₗᵢ[𝕜] Dual 𝕜 (Dual 𝕜 E) :=
   { inclusionInDoubleDual 𝕜 E with
     norm_map' := by

@@ -209,7 +209,7 @@ theorem smul_mem_polar {s : Set E} {x' : Dual 𝕜 E} {c : 𝕜} (hc : ∀ z, z
     rw [eq z]
     apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)
   have cancel : ‖c⁻¹‖ * ‖c‖ = 1 := by
-    simp only [c_zero, norm_eq_zero, Ne.def, not_false_iff, inv_mul_cancel, norm_inv]
+    simp only [c_zero, norm_eq_zero, Ne, not_false_iff, inv_mul_cancel, norm_inv]
   rwa [cancel] at le
 #align normed_space.smul_mem_polar NormedSpace.smul_mem_polar
 

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 -/
 import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
-import Mathlib.Analysis.NormedSpace.IsROrC
+import Mathlib.Analysis.NormedSpace.RCLike
 import Mathlib.Analysis.LocallyConvex.Polar
 
 #align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
@@ -107,7 +107,7 @@ end General
 
 section BidualIsometry
 
-variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable (𝕜 : Type v) [RCLike 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
     Compare `ContinuousLinearMap.opNorm_le_bound`. -/
@@ -118,7 +118,7 @@ theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual
     · simp only [h, hMp, norm_zero]
     · obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h
       calc
-        ‖x‖ = ‖(‖x‖ : 𝕜)‖ := IsROrC.norm_coe_norm.symm
+        ‖x‖ = ‖(‖x‖ : 𝕜)‖ := RCLike.norm_coe_norm.symm
         _ = ‖f x‖ := by rw [hfx]
         _ ≤ M * ‖f‖ := (hM f)
         _ = M := by rw [hf₁, mul_one]
@@ -240,7 +240,7 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
 
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
 inverse radius. -/
-theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
+theorem polar_closedBall {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
     (hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := by
   refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall 𝕜)
   intro x' h

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)

@@ -48,9 +48,7 @@ namespace NormedSpace
 section General
 
 variable (𝕜 : Type*) [NontriviallyNormedField 𝕜]
-
 variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
-
 variable (F : Type*) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 
 /-- The topological dual of a seminormed space `E`. -/
@@ -168,7 +166,6 @@ def polar (𝕜 : Type*) [NontriviallyNormedField 𝕜] {E : Type*} [SeminormedA
 #align normed_space.polar NormedSpace.polar
 
 variable (𝕜 : Type*) [NontriviallyNormedField 𝕜]
-
 variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 :=

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -38,7 +38,8 @@ dual
 
 noncomputable section
 
-open Classical Topology Bornology
+open scoped Classical
+open Topology Bornology
 
 universe u v
 

Co-authored-by: adomani <adomani@gmail.com>

@@ -77,7 +77,7 @@ theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f =
 
 theorem inclusionInDoubleDual_norm_eq :
     ‖inclusionInDoubleDual 𝕜 E‖ = ‖ContinuousLinearMap.id 𝕜 (Dual 𝕜 E)‖ :=
-  ContinuousLinearMap.op_norm_flip _
+  ContinuousLinearMap.opNorm_flip _
 #align normed_space.inclusion_in_double_dual_norm_eq NormedSpace.inclusionInDoubleDual_norm_eq
 
 theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by
@@ -86,7 +86,7 @@ theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1
 #align normed_space.inclusion_in_double_dual_norm_le NormedSpace.inclusionInDoubleDual_norm_le
 
 theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by
-  simpa using ContinuousLinearMap.le_of_op_norm_le _ (inclusionInDoubleDual_norm_le 𝕜 E) x
+  simpa using ContinuousLinearMap.le_of_opNorm_le _ (inclusionInDoubleDual_norm_le 𝕜 E) x
 #align normed_space.double_dual_bound NormedSpace.double_dual_bound
 
 /-- The dual pairing as a bilinear form. -/
@@ -111,7 +111,7 @@ section BidualIsometry
 variable (𝕜 : Type v) [IsROrC 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 /-- If one controls the norm of every `f x`, then one controls the norm of `x`.
-    Compare `ContinuousLinearMap.op_norm_le_bound`. -/
+    Compare `ContinuousLinearMap.opNorm_le_bound`. -/
 theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
     ‖x‖ ≤ M := by
   classical
@@ -221,7 +221,7 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
   rw [mem_polar_iff] at hx'
   simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *
   have hcr : 0 < ‖c‖ / r := div_pos (zero_lt_one.trans hc) hr
-  refine' ContinuousLinearMap.op_norm_le_of_shell hr hcr.le hc fun x h₁ h₂ => _
+  refine' ContinuousLinearMap.opNorm_le_of_shell hr hcr.le hc fun x h₁ h₂ => _
   calc
     ‖x' x‖ ≤ 1 := hx' _ h₂
     _ ≤ ‖c‖ / r * ‖x‖ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
@@ -232,7 +232,7 @@ variable (𝕜)
 theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
     closedBall (0 : Dual 𝕜 E) r⁻¹ ⊆ polar 𝕜 (closedBall (0 : E) r) := fun x' hx' x hx =>
   calc
-    ‖x' x‖ ≤ ‖x'‖ * ‖x‖ := x'.le_op_norm x
+    ‖x' x‖ ≤ ‖x'‖ * ‖x‖ := x'.le_opNorm x
     _ ≤ r⁻¹ * r :=
       (mul_le_mul (mem_closedBall_zero_iff.1 hx') (mem_closedBall_zero_iff.1 hx) (norm_nonneg _)
         (dist_nonneg.trans hx'))
@@ -247,7 +247,7 @@ theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [
   refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall 𝕜)
   intro x' h
   simp only [mem_closedBall_zero_iff]
-  refine' ContinuousLinearMap.op_norm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z _ => _
+  refine' ContinuousLinearMap.opNorm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z _ => _
   simpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.toLinearMap h z
 #align normed_space.polar_closed_ball NormedSpace.polar_closedBall
 

Use Bornology.IsBounded instead.

@@ -38,7 +38,7 @@ dual
 
 noncomputable section
 
-open Classical Topology
+open Classical Topology Bornology
 
 universe u v
 
@@ -225,7 +225,6 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
   calc
     ‖x' x‖ ≤ 1 := hx' _ h₂
     _ ≤ ‖c‖ / r * ‖x‖ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
-
 #align normed_space.polar_ball_subset_closed_ball_div NormedSpace.polar_ball_subset_closedBall_div
 
 variable (𝕜)
@@ -254,15 +253,14 @@ theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [
 
 /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
 of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
-theorem bounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
-    Bounded (polar 𝕜 s) := by
+theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
+    IsBounded (polar 𝕜 s) := by
   obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := NormedField.exists_one_lt_norm 𝕜
   obtain ⟨r, r_pos, r_ball⟩ : ∃ r : ℝ, 0 < r ∧ ball 0 r ⊆ s := Metric.mem_nhds_iff.1 s_nhd
-  exact
-    bounded_closedBall.mono
-      (((dualPairing 𝕜 E).flip.polar_antitone r_ball).trans <|
-        polar_ball_subset_closedBall_div ha r_pos)
-#align normed_space.bounded_polar_of_mem_nhds_zero NormedSpace.bounded_polar_of_mem_nhds_zero
+  exact isBounded_closedBall.subset
+    (((dualPairing 𝕜 E).flip.polar_antitone r_ball).trans <|
+      polar_ball_subset_closedBall_div ha r_pos)
+#align normed_space.bounded_polar_of_mem_nhds_zero NormedSpace.isBounded_polar_of_mem_nhds_zero
 
 end PolarSets
 

Following LinearAlgebra.Dual this makes NormedSpace.Dual reducible.

@@ -53,35 +53,16 @@ variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 variable (F : Type*) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 
 /-- The topological dual of a seminormed space `E`. -/
-def Dual :=
-  E →L[𝕜] 𝕜
+abbrev Dual : Type _ := E →L[𝕜] 𝕜
 #align normed_space.dual NormedSpace.Dual
 
--- Porting note: added manually
-section DerivedInstances
+-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_eq until
+-- leanprover/lean4#2522 is resolved; remove once fixed
+instance : NormedSpace 𝕜 (Dual 𝕜 E) := inferInstance
 
-instance : Inhabited (Dual 𝕜 E) :=
-  inferInstanceAs (Inhabited (E →L[𝕜] 𝕜))
-
-instance : SeminormedAddCommGroup (Dual 𝕜 E) :=
-  inferInstanceAs (SeminormedAddCommGroup (E →L[𝕜] 𝕜))
-
-instance : NormedSpace 𝕜 (Dual 𝕜 E) :=
-  inferInstanceAs (NormedSpace 𝕜 (E →L[𝕜] 𝕜))
-
-end DerivedInstances
-
-instance : ContinuousLinearMapClass (Dual 𝕜 E) 𝕜 E 𝕜 :=
-  ContinuousLinearMap.continuousSemilinearMapClass
-
-instance : CoeFun (Dual 𝕜 E) fun _ => E → 𝕜 :=
-  FunLike.hasCoeToFun
-
-instance : NormedAddCommGroup (Dual 𝕜 F) :=
-  ContinuousLinearMap.toNormedAddCommGroup
-
-instance [FiniteDimensional 𝕜 E] : FiniteDimensional 𝕜 (Dual 𝕜 E) :=
-  inferInstanceAs (FiniteDimensional 𝕜 (E →L[𝕜] 𝕜))
+-- TODO: helper instance for elaboration of inclusionInDoubleDual_norm_le until
+-- leanprover/lean4#2522 is resolved; remove once fixed
+instance : SeminormedAddCommGroup (Dual 𝕜 E) := inferInstance
 
 /-- The inclusion of a normed space in its double (topological) dual, considered
    as a bounded linear map. -/

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -46,11 +46,11 @@ namespace NormedSpace
 
 section General
 
-variable (𝕜 : Type _) [NontriviallyNormedField 𝕜]
+variable (𝕜 : Type*) [NontriviallyNormedField 𝕜]
 
-variable (E : Type _) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable (E : Type*) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
-variable (F : Type _) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+variable (F : Type*) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 
 /-- The topological dual of a seminormed space `E`. -/
 def Dual :=
@@ -180,14 +180,14 @@ open Metric Set NormedSpace
 /-- Given a subset `s` in a normed space `E` (over a field `𝕜`), the polar
 `polar 𝕜 s` is the subset of `Dual 𝕜 E` consisting of those functionals which
 evaluate to something of norm at most one at all points `z ∈ s`. -/
-def polar (𝕜 : Type _) [NontriviallyNormedField 𝕜] {E : Type _} [SeminormedAddCommGroup E]
+def polar (𝕜 : Type*) [NontriviallyNormedField 𝕜] {E : Type*} [SeminormedAddCommGroup E]
     [NormedSpace 𝕜 E] : Set E → Set (Dual 𝕜 E) :=
   (dualPairing 𝕜 E).flip.polar
 #align normed_space.polar NormedSpace.polar
 
-variable (𝕜 : Type _) [NontriviallyNormedField 𝕜]
+variable (𝕜 : Type*) [NontriviallyNormedField 𝕜]
 
-variable {E : Type _} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 :=
   Iff.rfl
@@ -262,7 +262,7 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
 
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with
 inverse radius. -/
-theorem polar_closedBall {𝕜 E : Type _} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
+theorem polar_closedBall {𝕜 E : Type*} [IsROrC 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
     (hr : 0 < r) : polar 𝕜 (closedBall (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := by
   refine' Subset.antisymm _ (closedBall_inv_subset_polar_closedBall 𝕜)
   intro x' h

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Heather Macbeth. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
-
-! This file was ported from Lean 3 source module analysis.normed_space.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.Analysis.NormedSpace.HahnBanach.Extension
 import Mathlib.Analysis.NormedSpace.IsROrC
 import Mathlib.Analysis.LocallyConvex.Polar
 
+#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
+
 /-!
 # The topological dual of a normed space
 

@@ -21,16 +21,16 @@ dual.
 
 For base field `𝕜 = ℝ` or `𝕜 = ℂ`, this map is actually an isometric embedding; we provide a
 version `NormedSpace.inclusionInDoubleDualLi` of the map which is of type a bundled linear
-isometric embedding, `E →ₗᵢ[𝕜] (dual 𝕜 (dual 𝕜 E))`.
+isometric embedding, `E →ₗᵢ[𝕜] (Dual 𝕜 (Dual 𝕜 E))`.
 
 Since a lot of elementary properties don't require `eq_of_dist_eq_zero` we start setting up the
 theory for `SeminormedAddCommGroup` and we specialize to `NormedAddCommGroup` when needed.
 
 ## Main definitions
 
-* `inclusion_in_double_dual` and `inclusion_in_double_dual_li` are the inclusion of a normed space
+* `inclusionInDoubleDual` and `inclusionInDoubleDualLi` are the inclusion of a normed space
   in its double dual, considered as a bounded linear map and as a linear isometry, respectively.
-* `polar 𝕜 s` is the subset of `dual 𝕜 E` consisting of those functionals `x'` for which
+* `polar 𝕜 s` is the subset of `Dual 𝕜 E` consisting of those functionals `x'` for which
   `‖x' z‖ ≤ 1` for every `z ∈ s`.
 
 ## Tags
@@ -181,7 +181,7 @@ section PolarSets
 open Metric Set NormedSpace
 
 /-- Given a subset `s` in a normed space `E` (over a field `𝕜`), the polar
-`polar 𝕜 s` is the subset of `dual 𝕜 E` consisting of those functionals which
+`polar 𝕜 s` is the subset of `Dual 𝕜 E` consisting of those functionals which
 evaluate to something of norm at most one at all points `z ∈ s`. -/
 def polar (𝕜 : Type _) [NontriviallyNormedField 𝕜] {E : Type _} [SeminormedAddCommGroup E]
     [NormedSpace 𝕜 E] : Set E → Set (Dual 𝕜 E) :=
@@ -261,7 +261,6 @@ theorem closedBall_inv_subset_polar_closedBall {r : ℝ} :
         (dist_nonneg.trans hx'))
     _ = r / r := (inv_mul_eq_div _ _)
     _ ≤ 1 := div_self_le_one r
-
 #align normed_space.closed_ball_inv_subset_polar_closed_ball NormedSpace.closedBall_inv_subset_polar_closedBall
 
 /-- The `polar` of closed ball in a normed space `E` is the closed ball of the dual with