Theory of topological modules and continuous linear maps.

We use the class ContinuousSMul for topological (semi) modules and topological vector spaces.

In this file we define continuous (semi-)linear maps, as semilinear maps between topological modules which are continuous. The set of continuous semilinear maps between the topological R₁-module M and R₂-module M₂ with respect to the RingHom σ is denoted by M →SL[σ] M₂. Plain linear maps are denoted by M →L[R] M₂ and star-linear maps by M →L⋆[R] M₂.

The corresponding notation for equivalences is M ≃SL[σ] M₂, M ≃L[R] M₂ and M ≃L⋆[R] M₂.

theorem ContinuousSMul.of_nhds_zero {R : Type u_1} {M : Type u_2} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [Module R M] [TopologicalRing R] [TopologicalAddGroup M] (hmul : Filter.Tendsto (fun p => p.fst • p.snd) (nhds 0 ×ˢ nhds 0) (nhds 0)) (hmulleft : ∀ (m : M), Filter.Tendsto (fun a => a • m) (nhds 0) (nhds 0)) (hmulright : ∀ (a : R), Filter.Tendsto (fun m => a • m) (nhds 0) (nhds 0)) : ContinuousSMul R M

theorem Submodule.eq_top_of_nonempty_interior' {R : Type u_1} {M : Type u_2} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] [Filter.NeBot (nhdsWithin 0 {x | IsUnit x})] (s : Submodule R M) (hs : Set.Nonempty (interior ↑s)) : s = ⊤

If M is a topological module over R and 0 is a limit of invertible elements of R, then ⊤ is the only submodule of M with a nonempty interior. This is the case, e.g., if R is a nontrivially normed field.

theorem Module.punctured_nhds_neBot (R : Type u_1) (M : Type u_2) [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] [Nontrivial M] [Filter.NeBot (nhdsWithin 0 {0}ᶜ)] [NoZeroSMulDivisors R M] (x : M) : Filter.NeBot (nhdsWithin x {x}ᶜ)

Let R be a topological ring such that zero is not an isolated point (e.g., a nontrivially normed field, see NormedField.punctured_nhds_neBot). Let M be a nontrivial module over R such that c • x = 0 implies c = 0 ∨ x = 0. Then M has no isolated points. We formulate this using NeBot (𝓝[≠] x).

This lemma is not an instance because Lean would need to find [ContinuousSMul ?m_1 M] with unknown ?m_1. We register this as an instance for R = ℝ in Real.punctured_nhds_module_neBot. One can also use haveI := Module.punctured_nhds_neBot R M in a proof.

theorem continuousSMul_induced {R : Type u_2} {M₁ : Type u_3} {M₂ : Type u_4} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] [u : TopologicalSpace R] {t : TopologicalSpace M₂} [ContinuousSMul R M₂] (f : M₁ →ₗ[R] M₂) : ContinuousSMul R M₁

theorem TopologicalSpace.IsSeparable.span {R : Type u_1} {M : Type u_2} [AddCommMonoid M] [Semiring R] [Module R M] [TopologicalSpace M] [TopologicalSpace R] [TopologicalSpace.SeparableSpace R] [ContinuousAdd M] [ContinuousSMul R M] {s : Set M} (hs : TopologicalSpace.IsSeparable s) : TopologicalSpace.IsSeparable ↑(Submodule.span R s)

The span of a separable subset with respect to a separable scalar ring is again separable.

instance Submodule.continuousSMul {α : Type u_1} {β : Type u_2} [TopologicalSpace β] [TopologicalSpace α] [Semiring α] [AddCommMonoid β] [Module α β] [ContinuousSMul α β] (S : Submodule α β) : ContinuousSMul α { x // x ∈ S }

instance Submodule.topologicalAddGroup {α : Type u_1} {β : Type u_2} [TopologicalSpace β] [Ring α] [AddCommGroup β] [Module α β] [TopologicalAddGroup β] (S : Submodule α β) : TopologicalAddGroup { x // x ∈ S }

theorem Submodule.mapsTo_smul_closure {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] (s : Submodule R M) (c : R) : Set.MapsTo (fun x => c • x) (closure ↑s) (closure ↑s)

theorem Submodule.smul_closure_subset {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] (s : Submodule R M) (c : R) : c • closure ↑s ⊆ closure ↑s

def Submodule.topologicalClosure {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : Submodule R M

The (topological-space) closure of a submodule of a topological R-module M is itself a submodule.

@[simp]

theorem Submodule.topologicalClosure_coe {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : ↑(Submodule.topologicalClosure s) = closure ↑s

theorem Submodule.le_topologicalClosure {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : s ≤ Submodule.topologicalClosure s

theorem Submodule.closure_subset_topologicalClosure_span {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Set M) : closure s ⊆ ↑(Submodule.topologicalClosure (Submodule.span R s))

theorem Submodule.isClosed_topologicalClosure {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) : IsClosed ↑(Submodule.topologicalClosure s)

theorem Submodule.topologicalClosure_minimal {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) {t : Submodule R M} (h : s ≤ t) (ht : IsClosed ↑t) : Submodule.topologicalClosure s ≤ t

theorem Submodule.topologicalClosure_mono {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s : Submodule R M} {t : Submodule R M} (h : s ≤ t) : Submodule.topologicalClosure s ≤ Submodule.topologicalClosure t

theorem IsClosed.submodule_topologicalClosure_eq {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s : Submodule R M} (hs : IsClosed ↑s) : Submodule.topologicalClosure s = s

The topological closure of a closed submodule s is equal to s.

theorem Submodule.dense_iff_topologicalClosure_eq_top {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s : Submodule R M} : Dense ↑s ↔ Submodule.topologicalClosure s = ⊤

A subspace is dense iff its topological closure is the entire space.

instance Submodule.topologicalClosure.completeSpace {R : Type u} [Semiring R] {M' : Type u_1} [AddCommMonoid M'] [Module R M'] [UniformSpace M'] [ContinuousAdd M'] [ContinuousConstSMul R M'] [CompleteSpace M'] (U : Submodule R M') : CompleteSpace { x // x ∈ Submodule.topologicalClosure U }

theorem Submodule.isClosed_or_dense_of_isCoatom {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) (hs : IsCoatom s) : IsClosed ↑s ∨ Dense ↑s

A maximal proper subspace of a topological module (i.e a Submodule satisfying IsCoatom) is either closed or dense.

theorem LinearMap.continuous_on_pi {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Finite ι] [Semiring R] [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous ↑f

structure ContinuousLinearMap {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (σ : R →+* S) (M : Type u_3) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_4) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends LinearMap : Type (max u_3 u_4)

• toFun : M → M₂
• map_add' : ∀ (x y : M), AddHom.toFun s.toAddHom (x + y) = AddHom.toFun s.toAddHom x + AddHom.toFun s.toAddHom y
• map_smul' : ∀ (r : R) (x : M), AddHom.toFun s.toAddHom (r • x) = ↑σ r • AddHom.toFun s.toAddHom x
• cont : Continuous s.toFun

  Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

def «term_→SL[_]_» : Lean.TrailingParserDescr

Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

def «term_→L[_]_» : Lean.TrailingParserDescr

Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

def «term_→L⋆[_]_» : Lean.TrailingParserDescr

Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological ring R.

class ContinuousSemilinearMapClass (F : Type u_1) {R : outParam (Type u_2)} {S : outParam (Type u_3)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) (M : outParam (Type u_4)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_5)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends SemilinearMapClass : Type (max (max u_1 u_4) u_5)

• coe : F → M → M₂
• coe_injective' : Function.Injective FunLike.coe
• map_add : ∀ (f : F) (x y : M), ↑f (x + y) = ↑f x + ↑f y
• map_smulₛₗ : ∀ (f : F) (r : R) (x : M), ↑f (r • x) = ↑σ r • ↑f x
• map_continuous : ∀ (f : F), Continuous ↑f

  Continuity

ContinuousSemilinearMapClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear maps M → M₂. See also ContinuousLinearMapClass F R M M₂ for the case where σ is the identity map on R. A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

@[inline, reducible]

abbrev ContinuousLinearMapClass (F : Type u_1) (R : outParam (Type u_2)) [Semiring R] (M : outParam (Type u_3)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_4)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Type (max (max u_1 u_3) u_4)

ContinuousLinearMapClass F R M M₂ asserts F is a type of bundled continuous R-linear maps M → M₂. This is an abbreviation for ContinuousSemilinearMapClass F (RingHom.id R) M M₂.

structure ContinuousLinearEquiv {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_3) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_4) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends LinearEquiv : Type (max u_3 u_4)

• toFun : M → M₂
• map_add' : ∀ (x y : M), AddHom.toFun s.toAddHom (x + y) = AddHom.toFun s.toAddHom x + AddHom.toFun s.toAddHom y
• map_smul' : ∀ (r : R) (x : M), AddHom.toFun s.toAddHom (r • x) = ↑σ r • AddHom.toFun s.toAddHom x
• invFun : M₂ → M
• left_inv : Function.LeftInverse s.invFun s.toFun
• right_inv : Function.RightInverse s.invFun s.toFun
• continuous_toFun : Continuous s.toFun

  Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

• continuous_invFun : Continuous s.invFun

  Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

def «term_≃SL[_]_» : Lean.TrailingParserDescr

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

def «term_≃L[_]_» : Lean.TrailingParserDescr

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

def «term_≃L⋆[_]_» : Lean.TrailingParserDescr

Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications M and M₂ will be topological modules over the topological semiring R.

class ContinuousSemilinearEquivClass (F : Type u_1) {R : outParam (Type u_2)} {S : outParam (Type u_3)} [Semiring R] [Semiring S] (σ : outParam (R →+* S)) {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam (Type u_4)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_5)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends SemilinearEquivClass : Type (max (max u_1 u_4) u_5)

• coe : F → M → M₂
• inv : F → M₂ → M
• left_inv : ∀ (e : F), Function.LeftInverse (EquivLike.inv e) (EquivLike.coe e)
• right_inv : ∀ (e : F), Function.RightInverse (EquivLike.inv e) (EquivLike.coe e)
• coe_injective' : ∀ (e g : F), EquivLike.coe e = EquivLike.coe g → EquivLike.inv e = EquivLike.inv g → e = g
• map_add : ∀ (f : F) (a b : M), ↑f (a + b) = ↑f a + ↑f b
• map_smulₛₗ : ∀ (f : F) (r : R) (x : M), ↑f (r • x) = ↑σ r • ↑f x
• map_continuous : ∀ (f : F), Continuous ↑f

  ContinuousSemilinearEquivClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear equivs M → M₂. See also ContinuousLinearEquivClass F R M M₂ for the case where σ is the identity map on R. A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

• inv_continuous : ∀ (f : F), Continuous (EquivLike.inv f)

  ContinuousSemilinearEquivClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear equivs M → M₂. See also ContinuousLinearEquivClass F R M M₂ for the case where σ is the identity map on R. A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

ContinuousSemilinearEquivClass F σ M M₂ asserts F is a type of bundled continuous σ-semilinear equivs M → M₂. See also ContinuousLinearEquivClass F R M M₂ for the case where σ is the identity map on R. A map f between an R-module and an S-module over a ring homomorphism σ : R →+* S is semilinear if it satisfies the two properties f (x + y) = f x + f y and f (c • x) = (σ c) • f x.

@[inline, reducible]

abbrev ContinuousLinearEquivClass (F : Type u_1) (R : outParam (Type u_2)) [Semiring R] (M : outParam (Type u_3)) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam (Type u_4)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Type (max (max u_1 u_3) u_4)

ContinuousLinearEquivClass F σ M M₂ asserts F is a type of bundled continuous R-linear equivs M → M₂. This is an abbreviation for ContinuousSemilinearEquivClass F (RingHom.id R) M M₂.

instance ContinuousSemilinearEquivClass.continuousSemilinearMapClass (F : Type u_1) {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] (σ : R →+* S) {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type u_4) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type u_5) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] [s : ContinuousSemilinearEquivClass F σ M M₂] : ContinuousSemilinearMapClass F σ M M₂

@[simp]

theorem linearMapOfMemClosureRangeCoe_apply {M₁ : Type u_1} {M₂ : Type u_2} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} (f : M₁ → M₂) (hf : f ∈ closure (Set.range FunLike.coe)) : ↑(linearMapOfMemClosureRangeCoe f hf) = (↑(addMonoidHomOfMemClosureRangeCoe f hf)).toFun

def linearMapOfMemClosureRangeCoe {M₁ : Type u_1} {M₂ : Type u_2} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} (f : M₁ → M₂) (hf : f ∈ closure (Set.range FunLike.coe)) : M₁ →ₛₗ[σ] M₂

Constructs a bundled linear map from a function and a proof that this function belongs to the closure of the set of linear maps.

@[simp]

theorem linearMapOfTendsto_apply {M₁ : Type u_1} {M₂ : Type u_2} {α : Type u_3} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [Filter.NeBot l] (h : Filter.Tendsto (fun a x => ↑(g a) x) l (nhds f)) : ↑(linearMapOfTendsto f g h) = f

def linearMapOfTendsto {M₁ : Type u_1} {M₂ : Type u_2} {α : Type u_3} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [Filter.NeBot l] (h : Filter.Tendsto (fun a x => ↑(g a) x) l (nhds f)) : M₁ →ₛₗ[σ] M₂

Construct a bundled linear map from a pointwise limit of linear maps

theorem LinearMap.isClosed_range_coe (M₁ : Type u_1) (M₂ : Type u_2) {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] (σ : R →+* S) : IsClosed (Set.range FunLike.coe)

Properties that hold for non-necessarily commutative semirings.

instance ContinuousLinearMap.LinearMap.coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Coe (M₁ →SL[σ₁₂] M₂) (M₁ →ₛₗ[σ₁₂] M₂)

Coerce continuous linear maps to linear maps.

theorem ContinuousLinearMap.coe_injective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Function.Injective ContinuousLinearMap.toLinearMap

instance ContinuousLinearMap.continuousSemilinearMapClass {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : ContinuousSemilinearMapClass (M₁ →SL[σ₁₂] M₂) σ₁₂ M₁ M₂

theorem ContinuousLinearMap.coe_mk {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) (h : Continuous f.toFun) : ↑(ContinuousLinearMap.mk f) = f

Coerce continuous linear maps to functions.

@[simp]

theorem ContinuousLinearMap.coe_mk' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) (h : Continuous f.toFun) : ↑(ContinuousLinearMap.mk f) = ↑f

theorem ContinuousLinearMap.continuous {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) : Continuous ↑f

theorem ContinuousLinearMap.uniformContinuous {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {E₁ : Type u_9} {E₂ : Type u_10} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (f : E₁ →SL[σ₁₂] E₂) : UniformContinuous ↑f

@[simp]

theorem ContinuousLinearMap.coe_inj {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} : ↑f = ↑g ↔ f = g

theorem ContinuousLinearMap.coeFn_injective {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Function.Injective FunLike.coe

def ContinuousLinearMap.Simps.apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (h : M₁ →SL[σ₁₂] M₂) : M₁ → M₂

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

def ContinuousLinearMap.Simps.coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (h : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂

See Note [custom simps projection].

theorem ContinuousLinearMap.ext {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} (h : ∀ (x : M₁), ↑f x = ↑g x) : f = g

theorem ContinuousLinearMap.ext_iff {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} : f = g ↔ ∀ (x : M₁), ↑f x = ↑g x

def ContinuousLinearMap.copy {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ↑f) : M₁ →SL[σ₁₂] M₂

Copy of a ContinuousLinearMap with a new toFun equal to the old one. Useful to fix definitional equalities.

@[simp]

theorem ContinuousLinearMap.coe_copy {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ↑f) : ↑(ContinuousLinearMap.copy f f' h) = f'

theorem ContinuousLinearMap.copy_eq {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ↑f) : ContinuousLinearMap.copy f f' h = f

theorem ContinuousLinearMap.map_zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) : ↑f 0 = 0

theorem ContinuousLinearMap.map_add {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (x : M₁) (y : M₁) : ↑f (x + y) = ↑f x + ↑f y

theorem ContinuousLinearMap.map_smulₛₗ {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (c : R₁) (x : M₁) : ↑f (c • x) = ↑σ₁₂ c • ↑f x

theorem ContinuousLinearMap.map_smul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (f : M₁ →L[R₁] M₂) (c : R₁) (x : M₁) : ↑f (c • x) = c • ↑f x

@[simp]

theorem ContinuousLinearMap.map_smul_of_tower {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_9} {S : Type u_10} [Semiring S] [SMul R M₁] [Module S M₁] [SMul R M₂] [Module S M₂] [LinearMap.CompatibleSMul M₁ M₂ R S] (f : M₁ →L[S] M₂) (c : R) (x : M₁) : ↑f (c • x) = c • ↑f x

@[deprecated map_sum]

theorem ContinuousLinearMap.map_sum {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {ι : Type u_9} (f : M₁ →SL[σ₁₂] M₂) (s : Finset ι) (g : ι → M₁) : ↑f (Finset.sum s fun i => g i) = Finset.sum s fun i => ↑f (g i)

@[simp]

theorem ContinuousLinearMap.coe_coe {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) : ↑↑f = ↑f

theorem ContinuousLinearMap.ext_ring {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] {f : R₁ →L[R₁] M₁} {g : R₁ →L[R₁] M₁} (h : ↑f 1 = ↑g 1) : f = g

theorem ContinuousLinearMap.ext_ring_iff {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] {f : R₁ →L[R₁] M₁} {g : R₁ →L[R₁] M₁} : f = g ↔ ↑f 1 = ↑g 1

theorem ContinuousLinearMap.eqOn_closure_span {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [T2Space M₂] {s : Set M₁} {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn (↑f) (↑g) s) : Set.EqOn (↑f) (↑g) (closure ↑(Submodule.span R₁ s))

If two continuous linear maps are equal on a set s, then they are equal on the closure of the Submodule.span of this set.

theorem ContinuousLinearMap.ext_on {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [T2Space M₂] {s : Set M₁} (hs : Dense ↑(Submodule.span R₁ s)) {f : M₁ →SL[σ₁₂] M₂} {g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn (↑f) (↑g) s) : f = g

If the submodule generated by a set s is dense in the ambient module, then two continuous linear maps equal on s are equal.

theorem Submodule.topologicalClosure_map {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (s : Submodule R₁ M₁) : Submodule.map (↑f) (Submodule.topologicalClosure s) ≤ Submodule.topologicalClosure (Submodule.map (↑f) s)

Under a continuous linear map, the image of the TopologicalClosure of a submodule is contained in the TopologicalClosure of its image.

theorem DenseRange.topologicalClosure_map_submodule {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [RingHomSurjective σ₁₂] [TopologicalSpace R₁] [TopologicalSpace R₂] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] [ContinuousSMul R₂ M₂] [ContinuousAdd M₂] {f : M₁ →SL[σ₁₂] M₂} (hf' : DenseRange ↑f) {s : Submodule R₁ M₁} (hs : Submodule.topologicalClosure s = ⊤) : Submodule.topologicalClosure (Submodule.map (↑f) s) = ⊤

Under a dense continuous linear map, a submodule whose TopologicalClosure is ⊤ is sent to another such submodule. That is, the image of a dense set under a map with dense range is dense.

instance ContinuousLinearMap.mulAction {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] : MulAction S₂ (M₁ →SL[σ₁₂] M₂)

theorem ContinuousLinearMap.smul_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : ↑(c • f) x = c • ↑f x

@[simp]

theorem ContinuousLinearMap.coe_smul {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : ↑(c • f) = c • ↑f

@[simp]

theorem ContinuousLinearMap.coe_smul' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} [Monoid S₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : ↑(c • f) = c • ↑f

instance ContinuousLinearMap.isScalarTower {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} {T₂ : Type u_10} [Monoid S₂] [Monoid T₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] [DistribMulAction T₂ M₂] [SMulCommClass R₂ T₂ M₂] [ContinuousConstSMul T₂ M₂] [SMul S₂ T₂] [IsScalarTower S₂ T₂ M₂] : IsScalarTower S₂ T₂ (M₁ →SL[σ₁₂] M₂)

instance ContinuousLinearMap.smulCommClass {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {S₂ : Type u_9} {T₂ : Type u_10} [Monoid S₂] [Monoid T₂] [DistribMulAction S₂ M₂] [SMulCommClass R₂ S₂ M₂] [ContinuousConstSMul S₂ M₂] [DistribMulAction T₂ M₂] [SMulCommClass R₂ T₂ M₂] [ContinuousConstSMul T₂ M₂] [SMulCommClass S₂ T₂ M₂] : SMulCommClass S₂ T₂ (M₁ →SL[σ₁₂] M₂)

instance ContinuousLinearMap.zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Zero (M₁ →SL[σ₁₂] M₂)

The continuous map that is constantly zero.

instance ContinuousLinearMap.inhabited {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : Inhabited (M₁ →SL[σ₁₂] M₂)

@[simp]

theorem ContinuousLinearMap.default_def {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : default = 0

@[simp]

theorem ContinuousLinearMap.zero_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (x : M₁) : ↑0 x = 0

@[simp]

theorem ContinuousLinearMap.coe_zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : ↑0 = 0

theorem ContinuousLinearMap.coe_zero' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] : ↑0 = 0

instance ContinuousLinearMap.uniqueOfLeft {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [Subsingleton M₁] : Unique (M₁ →SL[σ₁₂] M₂)

instance ContinuousLinearMap.uniqueOfRight {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [Subsingleton M₂] : Unique (M₁ →SL[σ₁₂] M₂)

theorem ContinuousLinearMap.exists_ne_zero {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {f : M₁ →SL[σ₁₂] M₂} (hf : f ≠ 0) : ∃ x, ↑f x ≠ 0

def ContinuousLinearMap.id (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : M₁ →L[R₁] M₁

the identity map as a continuous linear map.

instance ContinuousLinearMap.one {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : One (M₁ →L[R₁] M₁)

theorem ContinuousLinearMap.one_def {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : 1 = ContinuousLinearMap.id R₁ M₁

theorem ContinuousLinearMap.id_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (x : M₁) : ↑(ContinuousLinearMap.id R₁ M₁) x = x

@[simp]

theorem ContinuousLinearMap.coe_id {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : ↑(ContinuousLinearMap.id R₁ M₁) = LinearMap.id

@[simp]

theorem ContinuousLinearMap.coe_id' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : ↑(ContinuousLinearMap.id R₁ M₁) = id

@[simp]

theorem ContinuousLinearMap.coe_eq_id {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] {f : M₁ →L[R₁] M₁} : ↑f = LinearMap.id ↔ f = ContinuousLinearMap.id R₁ M₁

@[simp]

theorem ContinuousLinearMap.one_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (x : M₁) : ↑1 x = x

instance ContinuousLinearMap.add {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] : Add (M₁ →SL[σ₁₂] M₂)

@[simp]

theorem ContinuousLinearMap.add_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (g : M₁ →SL[σ₁₂] M₂) (x : M₁) : ↑(f + g) x = ↑f x + ↑g x

@[simp]

theorem ContinuousLinearMap.coe_add {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (g : M₁ →SL[σ₁₂] M₂) : ↑(f + g) = ↑f + ↑g

theorem ContinuousLinearMap.coe_add' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] (f : M₁ →SL[σ₁₂] M₂) (g : M₁ →SL[σ₁₂] M₂) : ↑(f + g) = ↑f + ↑g

instance ContinuousLinearMap.addCommMonoid {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] : AddCommMonoid (M₁ →SL[σ₁₂] M₂)

@[simp]

theorem ContinuousLinearMap.coe_sum {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) : ↑(Finset.sum t fun d => f d) = Finset.sum t fun d => ↑(f d)

@[simp]

theorem ContinuousLinearMap.coe_sum' {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) : ↑(Finset.sum t fun d => f d) = Finset.sum t fun d => ↑(f d)

theorem ContinuousLinearMap.sum_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] [ContinuousAdd M₂] {ι : Type u_9} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) (b : M₁) : ↑(Finset.sum t fun d => f d) b = Finset.sum t fun d => ↑(f d) b

def ContinuousLinearMap.comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : M₁ →SL[σ₁₃] M₃

Composition of bounded linear maps.

def ContinuousLinearMap.«term_∘L_» : Lean.TrailingParserDescr

Composition of bounded linear maps.

@[simp]

theorem ContinuousLinearMap.coe_comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : ↑(ContinuousLinearMap.comp h f) = LinearMap.comp ↑h ↑f

@[simp]

theorem ContinuousLinearMap.coe_comp' {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : ↑(ContinuousLinearMap.comp h f) = ↑h ∘ ↑f

theorem ContinuousLinearMap.comp_apply {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : ↑(ContinuousLinearMap.comp g f) x = ↑g (↑f x)

@[simp]

theorem ContinuousLinearMap.comp_id {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) : ContinuousLinearMap.comp f (ContinuousLinearMap.id R₁ M₁) = f

@[simp]

theorem ContinuousLinearMap.id_comp {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) : ContinuousLinearMap.comp (ContinuousLinearMap.id R₂ M₂) f = f

@[simp]

theorem ContinuousLinearMap.comp_zero {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (g : M₂ →SL[σ₂₃] M₃) : ContinuousLinearMap.comp g 0 = 0

@[simp]

theorem ContinuousLinearMap.zero_comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (f : M₁ →SL[σ₁₂] M₂) : ContinuousLinearMap.comp 0 f = 0

@[simp]

theorem ContinuousLinearMap.comp_add {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₁ →SL[σ₁₂] M₂) : ContinuousLinearMap.comp g (f₁ + f₂) = ContinuousLinearMap.comp g f₁ + ContinuousLinearMap.comp g f₂

@[simp]

theorem ContinuousLinearMap.add_comp {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [ContinuousAdd M₃] (g₁ : M₂ →SL[σ₂₃] M₃) (g₂ : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : ContinuousLinearMap.comp (g₁ + g₂) f = ContinuousLinearMap.comp g₁ f + ContinuousLinearMap.comp g₂ f

theorem ContinuousLinearMap.comp_assoc {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₂ M₂] [Module R₃ M₃] [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {R₄ : Type u_9} [Semiring R₄] [Module R₄ M₄] {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄} {σ₃₄ : R₃ →+* R₄} [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄) (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : ContinuousLinearMap.comp (ContinuousLinearMap.comp h g) f = ContinuousLinearMap.comp h (ContinuousLinearMap.comp g f)

instance ContinuousLinearMap.instMul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : Mul (M₁ →L[R₁] M₁)

theorem ContinuousLinearMap.mul_def {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (g : M₁ →L[R₁] M₁) : f * g = ContinuousLinearMap.comp f g

@[simp]

theorem ContinuousLinearMap.coe_mul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (g : M₁ →L[R₁] M₁) : ↑(f * g) = ↑f ∘ ↑g

theorem ContinuousLinearMap.mul_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (g : M₁ →L[R₁] M₁) (x : M₁) : ↑(f * g) x = ↑f (↑g x)

instance ContinuousLinearMap.monoidWithZero {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] : MonoidWithZero (M₁ →L[R₁] M₁)

theorem ContinuousLinearMap.coe_pow {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (f : M₁ →L[R₁] M₁) (n : ℕ) : ↑(f ^ n) = (↑f)^[n]

instance ContinuousLinearMap.semiring {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : Semiring (M₁ →L[R₁] M₁)

@[simp]

theorem ContinuousLinearMap.toLinearMapRingHom_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (self : M₁ →L[R₁] M₁) : ↑ContinuousLinearMap.toLinearMapRingHom self = ↑self

def ContinuousLinearMap.toLinearMapRingHom {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : (M₁ →L[R₁] M₁) →+* M₁ →ₗ[R₁] M₁

ContinuousLinearMap.toLinearMap as a RingHom.

instance ContinuousLinearMap.applyModule {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : Module (M₁ →L[R₁] M₁) M₁

The tautological action by M₁ →L[R₁] M₁ on M.

This generalizes Function.End.applyMulAction.

@[simp]

theorem ContinuousLinearMap.smul_def {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] (f : M₁ →L[R₁] M₁) (a : M₁) : f • a = ↑f a

instance ContinuousLinearMap.applyFaithfulSMul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : FaithfulSMul (M₁ →L[R₁] M₁) M₁

ContinuousLinearMap.applyModule is faithful.

instance ContinuousLinearMap.applySMulCommClass {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : SMulCommClass R₁ (M₁ →L[R₁] M₁) M₁

instance ContinuousLinearMap.applySMulCommClass' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : SMulCommClass (M₁ →L[R₁] M₁) R₁ M₁

instance ContinuousLinearMap.continuousConstSMul {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [ContinuousAdd M₁] : ContinuousConstSMul (M₁ →L[R₁] M₁) M₁

def ContinuousLinearMap.prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) : M₁ →L[R₁] M₂ × M₃

The cartesian product of two bounded linear maps, as a bounded linear map.

@[simp]

theorem ContinuousLinearMap.coe_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) : ↑(ContinuousLinearMap.prod f₁ f₂) = LinearMap.prod ↑f₁ ↑f₂

@[simp]

theorem ContinuousLinearMap.prod_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) (x : M₁) : ↑(ContinuousLinearMap.prod f₁ f₂) x = (↑f₁ x, ↑f₂ x)

def ContinuousLinearMap.inl (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : M₁ →L[R₁] M₁ × M₂

The left injection into a product is a continuous linear map.

def ContinuousLinearMap.inr (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : M₂ →L[R₁] M₁ × M₂

The right injection into a product is a continuous linear map.

@[simp]

theorem ContinuousLinearMap.inl_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (x : M₁) : ↑(ContinuousLinearMap.inl R₁ M₁ M₂) x = (x, 0)

@[simp]

theorem ContinuousLinearMap.inr_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] (x : M₂) : ↑(ContinuousLinearMap.inr R₁ M₁ M₂) x = (0, x)

@[simp]

theorem ContinuousLinearMap.coe_inl {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : ↑(ContinuousLinearMap.inl R₁ M₁ M₂) = LinearMap.inl R₁ M₁ M₂

@[simp]

theorem ContinuousLinearMap.coe_inr {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : ↑(ContinuousLinearMap.inr R₁ M₁ M₂) = LinearMap.inr R₁ M₁ M₂

theorem ContinuousLinearMap.isClosed_ker {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] {F : Type u_9} [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M₁ M₂] (f : F) : IsClosed ↑(LinearMap.ker f)

theorem ContinuousLinearMap.isComplete_ker {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {F : Type u_9} {M' : Type u_10} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) : IsComplete ↑(LinearMap.ker f)

instance ContinuousLinearMap.completeSpace_ker {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {F : Type u_9} {M' : Type u_10} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) : CompleteSpace { x // x ∈ LinearMap.ker f }

instance ContinuousLinearMap.completeSpace_eqLocus {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {F : Type u_9} {M' : Type u_10} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T2Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) (g : F) : CompleteSpace { x // x ∈ LinearMap.eqLocus f g }

@[simp]

theorem ContinuousLinearMap.ker_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) : LinearMap.ker (ContinuousLinearMap.prod f g) = LinearMap.ker f ⊓ LinearMap.ker g

def ContinuousLinearMap.codRestrict {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), ↑f x ∈ p) : M₁ →SL[σ₁₂] { x // x ∈ p }

Restrict codomain of a continuous linear map.

theorem ContinuousLinearMap.coe_codRestrict {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), ↑f x ∈ p) : ↑(ContinuousLinearMap.codRestrict f p h) = LinearMap.codRestrict p (↑f) h

@[simp]

theorem ContinuousLinearMap.coe_codRestrict_apply {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), ↑f x ∈ p) (x : M₁) : ↑(↑(ContinuousLinearMap.codRestrict f p h) x) = ↑f x

@[simp]

theorem ContinuousLinearMap.ker_codRestrict {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (x : M₁), ↑f x ∈ p) : LinearMap.ker (ContinuousLinearMap.codRestrict f p h) = LinearMap.ker f

def Submodule.subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) : { x // x ∈ p } →L[R₁] M₁

Submodule.subtype as a ContinuousLinearMap.

@[simp]

theorem Submodule.coe_subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) : ↑(Submodule.subtypeL p) = Submodule.subtype p

@[simp]

theorem Submodule.coe_subtypeL' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) : ↑(Submodule.subtypeL p) = ↑(Submodule.subtype p)

@[simp]

theorem Submodule.subtypeL_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) (x : { x // x ∈ p }) : ↑(Submodule.subtypeL p) x = ↑x

@[simp]

theorem Submodule.range_subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) : LinearMap.range (Submodule.subtypeL p) = p

@[simp]

theorem Submodule.ker_subtypeL {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] (p : Submodule R₁ M₁) : LinearMap.ker (Submodule.subtypeL p) = ⊥

def ContinuousLinearMap.fst (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : M₁ × M₂ →L[R₁] M₁

Prod.fst as a ContinuousLinearMap.

def ContinuousLinearMap.snd (R₁ : Type u_1) [Semiring R₁] (M₁ : Type u_4) [TopologicalSpace M₁] [AddCommMonoid M₁] (M₂ : Type u_6) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : M₁ × M₂ →L[R₁] M₂

Prod.snd as a ContinuousLinearMap.

@[simp]

theorem ContinuousLinearMap.coe_fst {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : ↑(ContinuousLinearMap.fst R₁ M₁ M₂) = LinearMap.fst R₁ M₁ M₂

@[simp]

theorem ContinuousLinearMap.coe_fst' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : ↑(ContinuousLinearMap.fst R₁ M₁ M₂) = Prod.fst

@[simp]

theorem ContinuousLinearMap.coe_snd {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : ↑(ContinuousLinearMap.snd R₁ M₁ M₂) = LinearMap.snd R₁ M₁ M₂

@[simp]

theorem ContinuousLinearMap.coe_snd' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : ↑(ContinuousLinearMap.snd R₁ M₁ M₂) = Prod.snd

@[simp]

theorem ContinuousLinearMap.fst_prod_snd {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₁ M₂] : ContinuousLinearMap.prod (ContinuousLinearMap.fst R₁ M₁ M₂) (ContinuousLinearMap.snd R₁ M₁ M₂) = ContinuousLinearMap.id R₁ (M₁ × M₂)

@[simp]

theorem ContinuousLinearMap.fst_comp_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) : ContinuousLinearMap.comp (ContinuousLinearMap.fst R₁ M₂ M₃) (ContinuousLinearMap.prod f g) = f

@[simp]

theorem ContinuousLinearMap.snd_comp_prod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) : ContinuousLinearMap.comp (ContinuousLinearMap.snd R₁ M₂ M₃) (ContinuousLinearMap.prod f g) = g

def ContinuousLinearMap.prodMap {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) : M₁ × M₃ →L[R₁] M₂ × M₄

Prod.map of two continuous linear maps.

@[simp]

theorem ContinuousLinearMap.coe_prodMap {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) : ↑(ContinuousLinearMap.prodMap f₁ f₂) = LinearMap.prodMap ↑f₁ ↑f₂

@[simp]

theorem ContinuousLinearMap.coe_prodMap' {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] {M₄ : Type u_8} [TopologicalSpace M₄] [AddCommMonoid M₄] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) : ↑(ContinuousLinearMap.prodMap f₁ f₂) = Prod.map ↑f₁ ↑f₂

def ContinuousLinearMap.coprod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) : M₁ × M₂ →L[R₁] M₃

The continuous linear map given by (x, y) ↦ f₁ x + f₂ y.

@[simp]

theorem ContinuousLinearMap.coe_coprod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) : ↑(ContinuousLinearMap.coprod f₁ f₂) = LinearMap.coprod ↑f₁ ↑f₂

@[simp]

theorem ContinuousLinearMap.coprod_apply {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) (x : M₁ × M₂) : ↑(ContinuousLinearMap.coprod f₁ f₂) x = ↑f₁ x.fst + ↑f₂ x.snd

theorem ContinuousLinearMap.range_coprod {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) : LinearMap.range (ContinuousLinearMap.coprod f₁ f₂) = LinearMap.range f₁ ⊔ LinearMap.range f₂

theorem ContinuousLinearMap.comp_fst_add_comp_snd {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {M₃ : Type u_7} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₁ M₁] [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f : M₁ →L[R₁] M₃) (g : M₂ →L[R₁] M₃) : ContinuousLinearMap.comp f (ContinuousLinearMap.fst R₁ M₁ M₂) + ContinuousLinearMap.comp g (ContinuousLinearMap.snd R₁ M₁ M₂) = ContinuousLinearMap.coprod f g

theorem ContinuousLinearMap.coprod_inl_inr {R₁ : Type u_1} [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M'₁ : Type u_5} [TopologicalSpace M'₁] [AddCommMonoid M'₁] [Module R₁ M₁] [Module R₁ M'₁] [ContinuousAdd M₁] [ContinuousAdd M'₁] : ContinuousLinearMap.coprod (ContinuousLinearMap.inl R₁ M₁ M'₁) (ContinuousLinearMap.inr R₁ M₁ M'₁) = ContinuousLinearMap.id R₁ (M₁ × M'₁)

def ContinuousLinearMap.smulRight {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_10} {S : Type u_11} [Semiring R] [Semiring S] [Module R M₁] [Module R M₂] [Module R S] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [ContinuousSMul S M₂] (c : M₁ →L[R] S) (f : M₂) : M₁ →L[R] M₂

The linear map fun x => c x • f. Associates to a scalar-valued linear map and an element of M₂ the M₂-valued linear map obtained by multiplying the two (a.k.a. tensoring by M₂). See also ContinuousLinearMap.smulRightₗ and ContinuousLinearMap.smulRightL.

@[simp]

theorem ContinuousLinearMap.smulRight_apply {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_10} {S : Type u_11} [Semiring R] [Semiring S] [Module R M₁] [Module R M₂] [Module R S] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [ContinuousSMul S M₂] {c : M₁ →L[R] S} {f : M₂} {x : M₁} : ↑(ContinuousLinearMap.smulRight c f) x = ↑c x • f

@[simp]

theorem ContinuousLinearMap.smulRight_one_one {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] (c : R₁ →L[R₁] M₂) : ContinuousLinearMap.smulRight 1 (↑c 1) = c

@[simp]

theorem ContinuousLinearMap.smulRight_one_eq_iff {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] {f : M₂} {f' : M₂} : ContinuousLinearMap.smulRight 1 f = ContinuousLinearMap.smulRight 1 f' ↔ f = f'

theorem ContinuousLinearMap.smulRight_comp {R₁ : Type u_1} [Semiring R₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] [ContinuousMul R₁] {x : M₂} {c : R₁} : ContinuousLinearMap.comp (ContinuousLinearMap.smulRight 1 x) (ContinuousLinearMap.smulRight 1 c) = ContinuousLinearMap.smulRight 1 (c • x)

def ContinuousLinearMap.toSpanSingleton (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] (x : M₁) : R₁ →L[R₁] M₁

Given an element x of a topological space M over a semiring R, the natural continuous linear map from R to M by taking multiples of x.

theorem ContinuousLinearMap.toSpanSingleton_apply (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] (x : M₁) (r : R₁) : ↑(ContinuousLinearMap.toSpanSingleton R₁ x) r = r • x

theorem ContinuousLinearMap.toSpanSingleton_add (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] [ContinuousAdd M₁] (x : M₁) (y : M₁) : ContinuousLinearMap.toSpanSingleton R₁ (x + y) = ContinuousLinearMap.toSpanSingleton R₁ x + ContinuousLinearMap.toSpanSingleton R₁ y

theorem ContinuousLinearMap.toSpanSingleton_smul' (R₁ : Type u_1) [Semiring R₁] {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] [Module R₁ M₁] [TopologicalSpace R₁] [ContinuousSMul R₁ M₁] {α : Type u_10} [Monoid α] [DistribMulAction α M₁] [ContinuousConstSMul α M₁] [SMulCommClass R₁ α M₁] (c : α) (x : M₁) : ContinuousLinearMap.toSpanSingleton R₁ (c • x) = c • ContinuousLinearMap.toSpanSingleton R₁ x

theorem ContinuousLinearMap.toSpanSingleton_smul (R : Type u_10) {M₁ : Type u_11} [CommSemiring R] [AddCommMonoid M₁] [Module R M₁] [TopologicalSpace R] [TopologicalSpace M₁] [ContinuousSMul R M₁] (c : R) (x : M₁) : ContinuousLinearMap.toSpanSingleton R (c • x) = c • ContinuousLinearMap.toSpanSingleton R x

A special case of to_span_singleton_smul' for when R is commutative.

def ContinuousLinearMap.pi {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) : M →L[R] (i : ι) → φ i

pi construction for continuous linear functions. From a family of continuous linear functions it produces a continuous linear function into a family of topological modules.

@[simp]

theorem ContinuousLinearMap.coe_pi' {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) : ↑(ContinuousLinearMap.pi f) = fun c i => ↑(f i) c

@[simp]

theorem ContinuousLinearMap.coe_pi {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) : ↑(ContinuousLinearMap.pi f) = LinearMap.pi fun i => ↑(f i)

theorem ContinuousLinearMap.pi_apply {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) (c : M) (i : ι) : ↑(ContinuousLinearMap.pi f) c i = ↑(f i) c

theorem ContinuousLinearMap.pi_eq_zero {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) : ContinuousLinearMap.pi f = 0 ↔ ∀ (i : ι), f i = 0

theorem ContinuousLinearMap.pi_zero {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] : (ContinuousLinearMap.pi fun x => 0) = 0

theorem ContinuousLinearMap.pi_comp {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M₂] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M →L[R] φ i) (g : M₂ →L[R] M) : ContinuousLinearMap.comp (ContinuousLinearMap.pi f) g = ContinuousLinearMap.pi fun i => ContinuousLinearMap.comp (f i) g

def ContinuousLinearMap.proj {R : Type u_1} [Semiring R] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (i : ι) : ((i : ι) → φ i) →L[R] φ i

The projections from a family of topological modules are continuous linear maps.

@[simp]

theorem ContinuousLinearMap.proj_apply {R : Type u_1} [Semiring R] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (i : ι) (b : (i : ι) → φ i) : ↑(ContinuousLinearMap.proj i) b = b i

theorem ContinuousLinearMap.proj_pi {R : Type u_1} [Semiring R] {M₂ : Type u_3} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M₂] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M₂ →L[R] φ i) (i : ι) : ContinuousLinearMap.comp (ContinuousLinearMap.proj i) (ContinuousLinearMap.pi f) = f i

theorem ContinuousLinearMap.iInf_ker_proj {R : Type u_1} [Semiring R] {ι : Type u_4} {φ : ι → Type u_5} [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] : ⨅ (i : ι), LinearMap.ker (ContinuousLinearMap.proj i) = ⊥

def ContinuousLinearMap.iInfKerProjEquiv (R : Type u_1) [Semiring R] {ι : Type u_4} (φ : ι → Type u_5) [(i : ι) → TopologicalSpace (φ i)] [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] {I : Set ι} {J : Set ι} [DecidablePred fun i => i ∈ I] (hd : Disjoint I J) (hu : Set.univ ⊆ I ∪ J) : { x // x ∈ ⨅ (i : ι) (_ : i ∈ J), LinearMap.ker (ContinuousLinearMap.proj i) } ≃L[R] (i : ↑I) → φ ↑i

If I and J are complementary index sets, the product of the kernels of the Jth projections of φ is linearly equivalent to the product over I.

theorem ContinuousLinearMap.map_neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (x : M) : ↑f (-x) = -↑f x

theorem ContinuousLinearMap.map_sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (x : M) (y : M) : ↑f (x - y) = ↑f x - ↑f y

@[simp]

theorem ContinuousLinearMap.sub_apply' {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) (x : M) : ↑(↑f - ↑g) x = ↑f x - ↑g x

theorem ContinuousLinearMap.range_prod_eq {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] {f : M →L[R] M₂} {g : M →L[R] M₃} (h : LinearMap.ker f ⊔ LinearMap.ker g = ⊤) : LinearMap.range (ContinuousLinearMap.prod f g) = Submodule.prod (LinearMap.range f) (LinearMap.range g)

theorem ContinuousLinearMap.ker_prod_ker_le_ker_coprod {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃) : Submodule.prod (LinearMap.ker f) (LinearMap.ker g) ≤ LinearMap.ker (ContinuousLinearMap.coprod f g)

theorem ContinuousLinearMap.ker_coprod_of_disjoint_range {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃) (hd : Disjoint (LinearMap.range f) (LinearMap.range g)) : LinearMap.ker (ContinuousLinearMap.coprod f g) = Submodule.prod (LinearMap.ker f) (LinearMap.ker g)

instance ContinuousLinearMap.neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] : Neg (M →SL[σ₁₂] M₂)

@[simp]

theorem ContinuousLinearMap.neg_apply {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (x : M) : ↑(-f) x = -↑f x

@[simp]

theorem ContinuousLinearMap.coe_neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) : ↑(-f) = -↑f

theorem ContinuousLinearMap.coe_neg' {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) : ↑(-f) = -↑f

instance ContinuousLinearMap.sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] : Sub (M →SL[σ₁₂] M₂)

instance ContinuousLinearMap.addCommGroup {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] : AddCommGroup (M →SL[σ₁₂] M₂)

theorem ContinuousLinearMap.sub_apply {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) (x : M) : ↑(f - g) x = ↑f x - ↑g x

@[simp]

theorem ContinuousLinearMap.coe_sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) : ↑(f - g) = ↑f - ↑g

@[simp]

theorem ContinuousLinearMap.coe_sub' {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [TopologicalAddGroup M₂] (f : M →SL[σ₁₂] M₂) (g : M →SL[σ₁₂] M₂) : ↑(f - g) = ↑f - ↑g

@[simp]

theorem ContinuousLinearMap.comp_neg {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : ContinuousLinearMap.comp g (-f) = -ContinuousLinearMap.comp g f

@[simp]

theorem ContinuousLinearMap.neg_comp {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : ContinuousLinearMap.comp (-g) f = -ContinuousLinearMap.comp g f

@[simp]

theorem ContinuousLinearMap.comp_sub {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ : M →SL[σ₁₂] M₂) (f₂ : M →SL[σ₁₂] M₂) : ContinuousLinearMap.comp g (f₁ - f₂) = ContinuousLinearMap.comp g f₁ - ContinuousLinearMap.comp g f₂

@[simp]

theorem ContinuousLinearMap.sub_comp {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {R₃ : Type u_3} [Ring R₃] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] {M₃ : Type u_6} [TopologicalSpace M₃] [AddCommGroup M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g₁ : M₂ →SL[σ₂₃] M₃) (g₂ : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : ContinuousLinearMap.comp (g₁ - g₂) f = ContinuousLinearMap.comp g₁ f - ContinuousLinearMap.comp g₂ f

instance ContinuousLinearMap.ring {R : Type u_1} [Ring R] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] [Module R M] [TopologicalAddGroup M] : Ring (M →L[R] M)

theorem ContinuousLinearMap.smulRight_one_pow {R : Type u_1} [Ring R] [TopologicalSpace R] [TopologicalRing R] (c : R) (n : ℕ) : ContinuousLinearMap.smulRight 1 c ^ n = ContinuousLinearMap.smulRight 1 (c ^ n)

def ContinuousLinearMap.projKerOfRightInverse {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse ↑f₂ ↑f₁) : M →L[R] { x // x ∈ LinearMap.ker f₁ }

Given a right inverse f₂ : M₂ →L[R] M to f₁ : M →L[R] M₂, projKerOfRightInverse f₁ f₂ h is the projection M →L[R] LinearMap.ker f₁ along LinearMap.range f₂.

@[simp]

theorem ContinuousLinearMap.coe_projKerOfRightInverse_apply {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse ↑f₂ ↑f₁) (x : M) : ↑(↑(ContinuousLinearMap.projKerOfRightInverse f₁ f₂ h) x) = x - ↑f₂ (↑f₁ x)

@[simp]

theorem ContinuousLinearMap.projKerOfRightInverse_apply_idem {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse ↑f₂ ↑f₁) (x : { x // x ∈ LinearMap.ker f₁ }) : ↑(ContinuousLinearMap.projKerOfRightInverse f₁ f₂ h) ↑x = x

@[simp]

theorem ContinuousLinearMap.projKerOfRightInverse_comp_inv {R : Type u_1} [Ring R] {R₂ : Type u_2} [Ring R₂] {M : Type u_4} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type u_5} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse ↑f₂ ↑f₁) (y : M₂) : ↑(ContinuousLinearMap.projKerOfRightInverse f₁ f₂ h) (↑f₂ y) = 0

theorem ContinuousLinearMap.isOpenMap_of_ne_zero {R : Type u_1} {M : Type u_2} [TopologicalSpace R] [DivisionRing R] [ContinuousSub R] [AddCommGroup M] [TopologicalSpace M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] (f : M →L[R] R) (hf : f ≠ 0) : IsOpenMap ↑f

A nonzero continuous linear functional is open.

@[simp]

theorem ContinuousLinearMap.smul_comp {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {S₃ : Type u_5} [Semiring R] [Semiring R₂] [Semiring R₃] [Monoid S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] [DistribMulAction S₃ M₃] [SMulCommClass R₃ S₃ M₃] [ContinuousConstSMul S₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : ContinuousLinearMap.comp (c • h) f = c • ContinuousLinearMap.comp h f

@[simp]

theorem ContinuousLinearMap.comp_smul {R : Type u_1} {S : Type u_4} [Semiring R] [Monoid S] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] [DistribMulAction S N₃] [SMulCommClass R S N₃] [ContinuousConstSMul S N₃] [DistribMulAction S N₂] [ContinuousConstSMul S N₂] [SMulCommClass R S N₂] [LinearMap.CompatibleSMul N₂ N₃ S R] (hₗ : N₂ →L[R] N₃) (c : S) (fₗ : M →L[R] N₂) : ContinuousLinearMap.comp hₗ (c • fₗ) = c • ContinuousLinearMap.comp hₗ fₗ

@[simp]

theorem ContinuousLinearMap.comp_smulₛₗ {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R] [Semiring R₂] [Semiring R₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {M₃ : Type u_8} [TopologicalSpace M₃] [AddCommMonoid M₃] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [SMulCommClass R₂ R₂ M₂] [SMulCommClass R₃ R₃ M₃] [ContinuousConstSMul R₂ M₂] [ContinuousConstSMul R₃ M₃] (h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂) : ContinuousLinearMap.comp h (c • f) = ↑σ₂₃ c • ContinuousLinearMap.comp h f

instance ContinuousLinearMap.distribMulAction {R : Type u_1} {R₂ : Type u_2} {S₃ : Type u_5} [Semiring R] [Semiring R₂] [Monoid S₃] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M₂ : Type u_7} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribMulAction S₃ M₂] [ContinuousConstSMul S₃ M₂] [SMulCommClass R₂ S₃ M₂] [ContinuousAdd M₂] : DistribMulAction S₃ (M →SL[σ₁₂] M₂)

@[simp]

theorem ContinuousLinearMap.prodEquiv_apply {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] (f : (M →L[R] N₂) × (M →L[R] N₃)) : ↑ContinuousLinearMap.prodEquiv f = ContinuousLinearMap.prod f.fst f.snd

def ContinuousLinearMap.prodEquiv {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] : (M →L[R] N₂) × (M →L[R] N₃) ≃ (M →L[R] N₂ × N₃)

ContinuousLinearMap.prod as an Equiv.

theorem ContinuousLinearMap.prod_ext_iff {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] {f : M × N₂ →L[R] N₃} {g : M × N₂ →L[R] N₃} : f = g ↔ ContinuousLinearMap.comp f (ContinuousLinearMap.inl R M N₂) = ContinuousLinearMap.comp g (ContinuousLinearMap.inl R M N₂) ∧ ContinuousLinearMap.comp f (ContinuousLinearMap.inr R M N₂) = ContinuousLinearMap.comp g (ContinuousLinearMap.inr R M N₂)

theorem ContinuousLinearMap.prod_ext {R : Type u_1} [Semiring R] {M : Type u_6} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {N₂ : Type u_9} [TopologicalSpace N₂] [AddCommMonoid N₂] [Module R N₂] {N₃ : Type u_10} [TopologicalSpace N₃] [AddCommMonoid N₃] [Module R N₃] {f : M × N₂ →L[R] N₃} {g : M × N₂ →L[R] N₃} (hl : ContinuousLinearMap.comp f (ContinuousLinearMap.inl R M N₂) = ContinuousLinearMap.comp g (ContinuousLinearMap.inl R M N₂)) (hr : ContinuousLinearMap.comp f (ContinuousLinearMap.inr R M N₂) = ContinuousLinearMap.comp g (ContinuousLinearMap.inr