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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ℝ) (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.

Equations

• LinearMap.mkContinuous f C h = { toLinearMap := f, cont := ⋯ }

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

Equations

• LinearMap.mkContinuousOfExistsBound f h = { toLinearMap := f, cont := ⋯ }

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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {f : E → F} (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 : ℝ} (h_bound : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) : Continuous f

theorem continuous_of_linear_of_bound {𝕜 : Type u_1} {E : Type u_3} {G : Type u_5} [Ring 𝕜] [SeminormedAddCommGroup E] [SeminormedAddCommGroup G] [Module 𝕜 E] [Module 𝕜 G] {f : E → G} (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 : ℝ} (h_bound : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) : Continuous f

@[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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) : ↑(LinearMap.mkContinuous f C h) = 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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) (x : E) : (LinearMap.mkContinuous f C h) 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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (h : ∃ (C : ℝ), ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) : ↑(LinearMap.mkContinuousOfExistsBound f h) = f

@[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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (h : ∃ (C : ℝ), ∀ (x : E), ‖f x‖ ≤ C * ‖x‖) (x : E) : (LinearMap.mkContinuousOfExistsBound f h) x = 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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (h : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖) : AntilipschitzWith K ⇑f

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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (h : AntilipschitzWith K ⇑f) (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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [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 : ℝ) (C_inv : ℝ) (h_to : ∀ (x : E), ‖e x‖ ≤ C_to * ‖x‖) (h_inv : ∀ (x : F), ‖(LinearEquiv.symm e) x‖ ≤ C_inv * ‖x‖) : E ≃SL[σ] F

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

Equations

• LinearEquiv.toContinuousLinearEquivOfBounds e C_to C_inv h_to h_inv = { toLinearEquiv := e, continuous_toFun := ⋯, continuous_invFun := ⋯ }

def LinearMap.toContinuousLinearMap₁ {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 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.

Equations

• LinearMap.toContinuousLinearMap₁ f = LinearMap.mkContinuous f ‖f 1‖ ⋯

@[simp]

theorem LinearMap.toContinuousLinearMap₁_coe {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : 𝕜 →ₗ[𝕜] E) : ↑(LinearMap.toContinuousLinearMap₁ f) = f

@[simp]

theorem LinearMap.toContinuousLinearMap₁_apply {𝕜 : Type u_1} {E : Type u_3} [SeminormedRing 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : 𝕜 →ₗ[𝕜] E) (x : 𝕜) : (LinearMap.toContinuousLinearMap₁ f) x = 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] [NormedAddCommGroup E] [NormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →SL[σ] F) {K : NNReal} (hf : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖) : UniformEmbedding ⇑f

Homotheties

def ContinuousLinearMap.ofHomothety {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} (f : E →ₛₗ[σ] F) (a : ℝ) (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.

Equations

• ContinuousLinearMap.ofHomothety f a hf = LinearMap.mkContinuous f a ⋯

theorem ContinuousLinearEquiv.homothety_inverse {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {F : Type u_4} [Ring 𝕜] [Ring 𝕜₂] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ σ₂₁] [RingHomInvPair σ₂₁ σ] (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) : (∀ (x : E), ‖f x‖ = a * ‖x‖) → ∀ (y : F), ‖(LinearEquiv.symm 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] [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F] {σ : 𝕜 →+* 𝕜₂} {σ₂₁ : 𝕜₂ →+* 𝕜} [RingHomInvPair σ σ₂₁] [RingHomInvPair σ₂₁ σ] (f : E ≃ₛₗ[σ] F) (a : ℝ) (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.

Equations

• ContinuousLinearEquiv.ofHomothety f a ha hf = LinearEquiv.toContinuousLinearEquivOfBounds f a a⁻¹ ⋯ ⋯