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Mathlib.Analysis.NormedSpace.ContinuousLinearMap

Constructions of continuous linear maps between (semi-)normed spaces #

A fundamental fact about (semi-)linear maps between normed spaces over sensible fields is that continuity and boundedness are equivalent conditions. That is, for normed spaces E, F, a LinearMap f : E →ₛₗ[σ] F is the coercion of some ContinuousLinearMap f' : E →SL[σ] F, if and only if there exists a bound C such that for all x, ‖f x‖ ≤ C * ‖x‖.

We prove one direction in this file: LinearMap.mkContinuous, boundedness implies continuity. The other direction, ContinuousLinearMap.bound, is deferred to a later file, where the strong operator topology on E →SL[σ] F is available, because it is natural to use ContinuousLinearMap.bound to define a norm ⨆ x, ‖f x‖ / ‖x‖ on E →SL[σ] F and to show that this is compatible with the strong operator topology.

This file also contains several corollaries of LinearMap.mkContinuous: other "easy" constructions of continuous linear maps between normed spaces.

This file is meant to be lightweight (it is imported by much of the analysis library); think twice before adding imports!

General constructions #

def LinearMap.mkContinuous {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ) (h : ∀ (x : E), f x C * x) :
E →SL[σ] F

Construct a continuous linear map from a linear map and a bound on this linear map. The fact that the norm of the continuous linear map is then controlled is given in LinearMap.mkContinuous_norm_le.

Instances For
def LinearMap.mkContinuousOfExistsBound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (h : C, ∀ (x : E), f x C * x) :
E →SL[σ] F

Construct a continuous linear map from a linear map and the existence of a bound on this linear map. If you have an explicit bound, use LinearMap.mkContinuous instead, as a norm estimate will follow automatically in LinearMap.mkContinuous_norm_le.

Instances For
theorem continuous_of_linear_of_boundₛₗ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {f : EF} (h_add : ∀ (x y : E), f (x + y) = f x + f y) (h_smul : ∀ (c : 𝕜) (x : E), f (c x) = σ c f x) {C : } (h_bound : ∀ (x : E), f x C * x) :
theorem continuous_of_linear_of_bound {𝕜 : Type u_1} {E : Type u_3} {G : Type u_5} [Ring 𝕜] [Module 𝕜 E] [Module 𝕜 G] {f : EG} (h_add : ∀ (x y : E), f (x + y) = f x + f y) (h_smul : ∀ (c : 𝕜) (x : E), f (c x) = c f x) {C : } (h_bound : ∀ (x : E), f x C * x) :
@[simp]
theorem LinearMap.mkContinuous_coe {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ) (h : ∀ (x : E), f x C * x) :
↑() = f
@[simp]
theorem LinearMap.mkContinuous_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ) (h : ∀ (x : E), f x C * x) (x : E) :
↑() x = f x
@[simp]
theorem LinearMap.mkContinuousOfExistsBound_coe {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (h : C, ∀ (x : E), f x C * x) :
@[simp]
theorem LinearMap.mkContinuousOfExistsBound_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (h : C, ∀ (x : E), f x C * x) (x : E) :
= f x
theorem ContinuousLinearMap.antilipschitz_of_bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (h : ∀ (x : E), x K * f x) :
theorem ContinuousLinearMap.bound_of_antilipschitz {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (h : ) (x : E) :
x K * f x
def LinearEquiv.toContinuousLinearEquivOfBounds {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ σ₂₁] [RingHomInvPair σ₂₁ σ] (e : E ≃ₛₗ[σ] F) (C_to : ) (C_inv : ) (h_to : ∀ (x : E), e x C_to * x) (h_inv : ∀ (x : F), ↑() x C_inv * x) :
E ≃SL[σ] F

Construct a continuous linear equivalence from a linear equivalence together with bounds in both directions.

Instances For
def LinearMap.toContinuousLinearMap₁ {𝕜 : Type u_1} {E : Type u_3} [] [Module 𝕜 E] [] (f : 𝕜 →ₗ[𝕜] E) :
𝕜 →L[𝕜] E

Reinterpret a linear map 𝕜 →ₗ[𝕜] E as a continuous linear map. This construction is generalized to the case of any finite dimensional domain in LinearMap.toContinuousLinearMap.

Instances For
@[simp]
theorem LinearMap.toContinuousLinearMap₁_coe {𝕜 : Type u_1} {E : Type u_3} [] [Module 𝕜 E] [] (f : 𝕜 →ₗ[𝕜] E) :
@[simp]
theorem LinearMap.toContinuousLinearMap₁_apply {𝕜 : Type u_1} {E : Type u_3} [] [Module 𝕜 E] [] (f : 𝕜 →ₗ[𝕜] E) (x : 𝕜) :
= f x
theorem ContinuousLinearMap.uniformEmbedding_of_bound {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (hf : ∀ (x : E), x K * f x) :

Homotheties #

def ContinuousLinearMap.ofHomothety {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (a : ) (hf : ∀ (x : E), f x = a * x) :
E →SL[σ] F

A (semi-)linear map which is a homothety is a continuous linear map. Since the field 𝕜 need not have ℝ as a subfield, this theorem is not directly deducible from the corresponding theorem about isometries plus a theorem about scalar multiplication. Likewise for the other theorems about homotheties in this file.

Instances For
theorem ContinuousLinearEquiv.homothety_inverse {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ σ₂₁] [RingHomInvPair σ₂₁ σ] (a : ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
(∀ (x : E), f x = a * x) → ∀ (y : F), ↑() y = a⁻¹ * y
noncomputable def ContinuousLinearEquiv.ofHomothety {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ σ₂₁] [RingHomInvPair σ₂₁ σ] (f : E ≃ₛₗ[σ] F) (a : ) (ha : 0 < a) (hf : ∀ (x : E), f x = a * x) :
E ≃SL[σ] F

A linear equivalence which is a homothety is a continuous linear equivalence.

Instances For

The span of a single vector #

theorem ContinuousLinearMap.toSpanSingleton_homothety (𝕜 : Type u_1) {E : Type u_3} [Module 𝕜 E] [] (x : E) (c : 𝕜) :
theorem ContinuousLinearEquiv.toSpanNonzeroSingleton_homothety (𝕜 : Type u_1) {E : Type u_3} [Module 𝕜 E] [] (x : E) (h : x 0) (c : 𝕜) :
noncomputable def ContinuousLinearEquiv.toSpanNonzeroSingleton (𝕜 : Type u_1) {E : Type u_3} [] [] (x : E) (h : x 0) :
𝕜 ≃L[𝕜] { x // x Submodule.span 𝕜 {x} }

Given a nonzero element x of a normed space E₁ over a field 𝕜, the natural continuous linear equivalence from E₁ to the span of x.

Instances For
noncomputable def ContinuousLinearEquiv.coord (𝕜 : Type u_1) {E : Type u_3} [] [] (x : E) (h : x 0) :
{ x // x Submodule.span 𝕜 {x} } →L[𝕜] 𝕜

Given a nonzero element x of a normed space E₁ over a field 𝕜, the natural continuous linear map from the span of x to 𝕜.

Instances For
@[simp]
theorem ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm (𝕜 : Type u_1) {E : Type u_3} [] [] {x : E} (h : x 0) :
@[simp]
theorem ContinuousLinearEquiv.coord_toSpanNonzeroSingleton (𝕜 : Type u_1) {E : Type u_3} [] [] {x : E} (h : x 0) (c : 𝕜) :
↑() () = c
@[simp]
theorem ContinuousLinearEquiv.toSpanNonzeroSingleton_coord (𝕜 : Type u_1) {E : Type u_3} [] [] {x : E} (h : x 0) (y : { x // x Submodule.span 𝕜 {x} }) :
↑() (↑() y) = y
@[simp]
theorem ContinuousLinearEquiv.coord_self (𝕜 : Type u_1) {E : Type u_3} [] [] (x : E) (h : x 0) :
↑() { val := x, property := (_ : x Submodule.span 𝕜 {x}) } = 1