Operator norm: bilinear maps

This file contains lemmas concerning operator norm as applied to bilinear maps E × F → G, interpreted as linear maps E → F → G as usual (and similarly for semilinear variants).

theorem ContinuousLinearMap.opNorm_ext {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G) (h : ∀ (x : E), ‖f x‖ = ‖g x‖) : ‖f‖ = ‖g‖

@[deprecated ContinuousLinearMap.opNorm_ext]

theorem ContinuousLinearMap.op_norm_ext {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F) (g : E →SL[σ₁₃] G) (h : ∀ (x : E), ‖f x‖ = ‖g x‖) : ‖f‖ = ‖g‖

Alias of ContinuousLinearMap.opNorm_ext.

theorem ContinuousLinearMap.opNorm_le_bound₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C) (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) : ‖f‖ ≤ C

@[deprecated ContinuousLinearMap.opNorm_le_bound₂]

theorem ContinuousLinearMap.op_norm_le_bound₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C) (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) : ‖f‖ ≤ C

Alias of ContinuousLinearMap.opNorm_le_bound₂.

theorem ContinuousLinearMap.le_opNorm₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) : ‖(f x) y‖ ≤ ‖f‖ * ‖x‖ * ‖y‖

@[deprecated ContinuousLinearMap.le_opNorm₂]

theorem ContinuousLinearMap.le_op_norm₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) : ‖(f x) y‖ ≤ ‖f‖ * ‖x‖ * ‖y‖

Alias of ContinuousLinearMap.le_opNorm₂.

theorem ContinuousLinearMap.le_of_opNorm₂_le_of_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {x : E} {y : F} {a : ℝ} {b : ℝ} {c : ℝ} (hf : ‖f‖ ≤ a) (hx : ‖x‖ ≤ b) (hy : ‖y‖ ≤ c) : ‖(f x) y‖ ≤ a * b * c

@[deprecated ContinuousLinearMap.le_of_opNorm₂_le_of_le]

theorem ContinuousLinearMap.le_of_op_norm₂_le_of_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) {x : E} {y : F} {a : ℝ} {b : ℝ} {c : ℝ} (hf : ‖f‖ ≤ a) (hx : ‖x‖ ≤ b) (hy : ‖y‖ ≤ c) : ‖(f x) y‖ ≤ a * b * c

Alias of ContinuousLinearMap.le_of_opNorm₂_le_of_le.

theorem LinearMap.norm_mkContinuous₂_aux {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ℝ) (h : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) (x : E) : ‖(f x).mkContinuous (C * ‖x‖) ⋯‖ ≤ (C ⊔ 0) * ‖x‖

noncomputable def LinearMap.mkContinuousOfExistsBound₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (h : ∃ (C : ℝ), ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) : E →SL[σ₁₃] F →SL[σ₂₃] G

Create a bilinear map (represented as a map E →L[𝕜] F →L[𝕜] G) from the corresponding linear map and existence of a bound on the norm of the image. The linear map can be constructed using LinearMap.mk₂.

If you have an explicit bound, use LinearMap.mkContinuous₂ instead, as a norm estimate will follow automatically in LinearMap.mkContinuous₂_norm_le.

Equations

• f.mkContinuousOfExistsBound₂ h = { toFun := fun (x : E) => (f x).mkContinuousOfExistsBound ⋯, map_add' := ⋯, map_smul' := ⋯ }.mkContinuousOfExistsBound ⋯

noncomputable def LinearMap.mkContinuous₂ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ℝ) (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) : E →SL[σ₁₃] F →SL[σ₂₃] G

Create a bilinear map (represented as a map E →L[𝕜] F →L[𝕜] G) from the corresponding linear map and a bound on the norm of the image. The linear map can be constructed using LinearMap.mk₂. Lemmas LinearMap.mkContinuous₂_norm_le' and LinearMap.mkContinuous₂_norm_le provide estimates on the norm of an operator constructed using this function.

Equations

• f.mkContinuous₂ C hC = f.mkContinuousOfExistsBound₂ ⋯

@[simp]

theorem LinearMap.mkContinuous₂_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) (x : E) (y : F) : ((f.mkContinuous₂ C hC) x) y = (f x) y

theorem LinearMap.mkContinuous₂_norm_le' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) : ‖f.mkContinuous₂ C hC‖ ≤ C ⊔ 0

theorem LinearMap.mkContinuous₂_norm_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) {C : ℝ} (h0 : 0 ≤ C) (hC : ∀ (x : E) (y : F), ‖(f x) y‖ ≤ C * ‖x‖ * ‖y‖) : ‖f.mkContinuous₂ C hC‖ ≤ C

noncomputable def ContinuousLinearMap.flip {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : F →SL[σ₂₃] E →SL[σ₁₃] G

Flip the order of arguments of a continuous bilinear map. For a version bundled as LinearIsometryEquiv, see ContinuousLinearMap.flipL.

Equations

• f.flip = (LinearMap.mk₂'ₛₗ σ₂₃ σ₁₃ (fun (y : F) (x : E) => (f x) y) ⋯ ⋯ ⋯ ⋯).mkContinuous₂ ‖f‖ ⋯

@[simp]

theorem ContinuousLinearMap.flip_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (x : E) (y : F) : (f.flip y) x = (f x) y

@[simp]

theorem ContinuousLinearMap.flip_flip {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : f.flip.flip = f

@[simp]

theorem ContinuousLinearMap.opNorm_flip {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : ‖f.flip‖ = ‖f‖

@[deprecated ContinuousLinearMap.opNorm_flip]

theorem ContinuousLinearMap.op_norm_flip {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : ‖f.flip‖ = ‖f‖

Alias of ContinuousLinearMap.opNorm_flip.

@[simp]

theorem ContinuousLinearMap.flip_add {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (g : E →SL[σ₁₃] F →SL[σ₂₃] G) : (f + g).flip = f.flip + g.flip

@[simp]

theorem ContinuousLinearMap.flip_smul {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] (c : 𝕜₃) (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : (c • f).flip = c • f.flip

noncomputable def ContinuousLinearMap.flipₗᵢ' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} (E : Type u_4) (F : Type u_6) (G : Type u_8) [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] (σ₂₃ : 𝕜₂ →+* 𝕜₃) (σ₁₃ : 𝕜 →+* 𝕜₃) [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] : (E →SL[σ₁₃] F →SL[σ₂₃] G) ≃ₗᵢ[𝕜₃] F →SL[σ₂₃] E →SL[σ₁₃] G

Flip the order of arguments of a continuous bilinear map. This is a version bundled as a LinearIsometryEquiv. For an unbundled version see ContinuousLinearMap.flip.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem ContinuousLinearMap.flipₗᵢ'_symm {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] : (ContinuousLinearMap.flipₗᵢ' E F G σ₂₃ σ₁₃).symm = ContinuousLinearMap.flipₗᵢ' F E G σ₁₃ σ₂₃

@[simp]

theorem ContinuousLinearMap.coe_flipₗᵢ' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] : ⇑(ContinuousLinearMap.flipₗᵢ' E F G σ₂₃ σ₁₃) = ContinuousLinearMap.flip

noncomputable def ContinuousLinearMap.flipₗᵢ (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] : (E →L[𝕜] Fₗ →L[𝕜] Gₗ) ≃ₗᵢ[𝕜] Fₗ →L[𝕜] E →L[𝕜] Gₗ

Flip the order of arguments of a continuous bilinear map. This is a version bundled as a LinearIsometryEquiv. For an unbundled version see ContinuousLinearMap.flip.

Equations

• ContinuousLinearMap.flipₗᵢ 𝕜 E Fₗ Gₗ = { toFun := ContinuousLinearMap.flip, map_add' := ⋯, map_smul' := ⋯, invFun := ContinuousLinearMap.flip, left_inv := ⋯, right_inv := ⋯, norm_map' := ⋯ }

@[simp]

theorem ContinuousLinearMap.flipₗᵢ_symm {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] : (ContinuousLinearMap.flipₗᵢ 𝕜 E Fₗ Gₗ).symm = ContinuousLinearMap.flipₗᵢ 𝕜 Fₗ E Gₗ

@[simp]

theorem ContinuousLinearMap.coe_flipₗᵢ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] : ⇑(ContinuousLinearMap.flipₗᵢ 𝕜 E Fₗ Gₗ) = ContinuousLinearMap.flip

noncomputable def ContinuousLinearMap.apply' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} (F : Type u_6) [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] (σ₁₂ : 𝕜 →+* 𝕜₂) [RingHomIsometric σ₁₂] : E →SL[σ₁₂] (E →SL[σ₁₂] F) →L[𝕜₂] F

The continuous semilinear map obtained by applying a continuous semilinear map at a given vector.

This is the continuous version of LinearMap.applyₗ.

Equations

• ContinuousLinearMap.apply' F σ₁₂ = (ContinuousLinearMap.id 𝕜₂ (E →SL[σ₁₂] F)).flip

@[simp]

theorem ContinuousLinearMap.apply_apply' {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_6} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (v : E) (f : E →SL[σ₁₂] F) : ((ContinuousLinearMap.apply' F σ₁₂) v) f = f v

noncomputable def ContinuousLinearMap.apply (𝕜 : Type u_1) {E : Type u_4} (Fₗ : Type u_7) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] : E →L[𝕜] (E →L[𝕜] Fₗ) →L[𝕜] Fₗ

The continuous semilinear map obtained by applying a continuous semilinear map at a given vector.

This is the continuous version of LinearMap.applyₗ.

Equations

• ContinuousLinearMap.apply 𝕜 Fₗ = (ContinuousLinearMap.id 𝕜 (E →L[𝕜] Fₗ)).flip

@[simp]

theorem ContinuousLinearMap.apply_apply {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (v : E) (f : E →L[𝕜] Fₗ) : ((ContinuousLinearMap.apply 𝕜 Fₗ) v) f = f v

noncomputable def ContinuousLinearMap.compSL {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} (E : Type u_4) (F : Type u_6) (G : Type u_8) [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] (σ₁₂ : 𝕜 →+* 𝕜₂) (σ₂₃ : 𝕜₂ →+* 𝕜₃) {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] : (F →SL[σ₂₃] G) →L[𝕜₃] (E →SL[σ₁₂] F) →SL[σ₂₃] E →SL[σ₁₃] G

Composition of continuous semilinear maps as a continuous semibilinear map.

Equations

• ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃ = (LinearMap.mk₂'ₛₗ (RingHom.id 𝕜₃) σ₂₃ ContinuousLinearMap.comp ⋯ ⋯ ⋯ ⋯).mkContinuous₂ 1 ⋯

theorem ContinuousLinearMap.norm_compSL_le {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} (E : Type u_4) (F : Type u_6) (G : Type u_8) [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] (σ₁₂ : 𝕜 →+* 𝕜₂) (σ₂₃ : 𝕜₂ →+* 𝕜₃) {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] : ‖ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃‖ ≤ 1

@[simp]

theorem ContinuousLinearMap.compSL_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] (f : F →SL[σ₂₃] G) (g : E →SL[σ₁₂] F) : ((ContinuousLinearMap.compSL E F G σ₁₂ σ₂₃) f) g = f.comp g

theorem Continuous.const_clm_comp {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] {X : Type u_11} [TopologicalSpace X] {f : X → E →SL[σ₁₂] F} (hf : Continuous f) (g : F →SL[σ₂₃] G) : Continuous fun (x : X) => g.comp (f x)

theorem Continuous.clm_comp_const {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₁₂ : 𝕜 →+* 𝕜₂} {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃] [RingHomIsometric σ₁₂] {X : Type u_11} [TopologicalSpace X] {g : X → F →SL[σ₂₃] G} (hg : Continuous g) (f : E →SL[σ₁₂] F) : Continuous fun (x : X) => (g x).comp f

noncomputable def ContinuousLinearMap.compL (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] : (Fₗ →L[𝕜] Gₗ) →L[𝕜] (E →L[𝕜] Fₗ) →L[𝕜] E →L[𝕜] Gₗ

Composition of continuous linear maps as a continuous bilinear map.

Equations

• ContinuousLinearMap.compL 𝕜 E Fₗ Gₗ = ContinuousLinearMap.compSL E Fₗ Gₗ (RingHom.id 𝕜) (RingHom.id 𝕜)

theorem ContinuousLinearMap.norm_compL_le (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] : ‖ContinuousLinearMap.compL 𝕜 E Fₗ Gₗ‖ ≤ 1

@[simp]

theorem ContinuousLinearMap.compL_apply (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) (Gₗ : Type u_9) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : Fₗ →L[𝕜] Gₗ) (g : E →L[𝕜] Fₗ) : ((ContinuousLinearMap.compL 𝕜 E Fₗ Gₗ) f) g = f.comp g

noncomputable def ContinuousLinearMap.precompR {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : E →L[𝕜] (Eₗ →L[𝕜] Fₗ) →L[𝕜] Eₗ →L[𝕜] Gₗ

Apply L(x,-) pointwise to bilinear maps, as a continuous bilinear map

Equations

• ContinuousLinearMap.precompR Eₗ L = (ContinuousLinearMap.compL 𝕜 Eₗ Fₗ Gₗ).comp L

@[simp]

theorem ContinuousLinearMap.precompR_apply {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : ∀ (a : E), (ContinuousLinearMap.precompR Eₗ L) a = (ContinuousLinearMap.compL 𝕜 Eₗ Fₗ Gₗ) (L a)

noncomputable def ContinuousLinearMap.precompL {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : (Eₗ →L[𝕜] E) →L[𝕜] Fₗ →L[𝕜] Eₗ →L[𝕜] Gₗ

Apply L(-,y) pointwise to bilinear maps, as a continuous bilinear map

Equations

• ContinuousLinearMap.precompL Eₗ L = (ContinuousLinearMap.precompR Eₗ L.flip).flip

@[simp]

theorem ContinuousLinearMap.precompL_apply {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (u : Eₗ →L[𝕜] E) (f : Fₗ) (g : Eₗ) : (((ContinuousLinearMap.precompL Eₗ L) u) f) g = (L (u g)) f

theorem ContinuousLinearMap.norm_precompR_le {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : ‖ContinuousLinearMap.precompR Eₗ L‖ ≤ ‖L‖

theorem ContinuousLinearMap.norm_precompL_le {𝕜 : Type u_1} {E : Type u_4} (Eₗ : Type u_5) {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Eₗ] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Eₗ] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (L : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : ‖ContinuousLinearMap.precompL Eₗ L‖ ≤ ‖L‖

noncomputable def ContinuousLinearMap.bilinearComp {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {E' : Type u_11} {F' : Type u_12} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {𝕜₁' : Type u_13} {𝕜₂' : Type u_14} [NontriviallyNormedField 𝕜₁'] [NontriviallyNormedField 𝕜₂'] [NormedSpace 𝕜₁' E'] [NormedSpace 𝕜₂' F'] {σ₁' : 𝕜₁' →+* 𝕜} {σ₁₃' : 𝕜₁' →+* 𝕜₃} {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [RingHomCompTriple σ₁' σ₁₃ σ₁₃'] [RingHomCompTriple σ₂' σ₂₃ σ₂₃'] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃'] [RingHomIsometric σ₂₃'] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) : E' →SL[σ₁₃'] F' →SL[σ₂₃'] G

Compose a bilinear map E →SL[σ₁₃] F →SL[σ₂₃] G with two linear maps E' →SL[σ₁'] E and F' →SL[σ₂'] F.

Equations

• f.bilinearComp gE gF = ((f.comp gE).flip.comp gF).flip

@[simp]

theorem ContinuousLinearMap.bilinearComp_apply {𝕜 : Type u_1} {𝕜₂ : Type u_2} {𝕜₃ : Type u_3} {E : Type u_4} {F : Type u_6} {G : Type u_8} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [SeminormedAddCommGroup G] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NontriviallyNormedField 𝕜₃] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [NormedSpace 𝕜₃ G] {σ₂₃ : 𝕜₂ →+* 𝕜₃} {σ₁₃ : 𝕜 →+* 𝕜₃} {E' : Type u_11} {F' : Type u_12} [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] {𝕜₁' : Type u_13} {𝕜₂' : Type u_14} [NontriviallyNormedField 𝕜₁'] [NontriviallyNormedField 𝕜₂'] [NormedSpace 𝕜₁' E'] [NormedSpace 𝕜₂' F'] {σ₁' : 𝕜₁' →+* 𝕜} {σ₁₃' : 𝕜₁' →+* 𝕜₃} {σ₂' : 𝕜₂' →+* 𝕜₂} {σ₂₃' : 𝕜₂' →+* 𝕜₃} [RingHomCompTriple σ₁' σ₁₃ σ₁₃'] [RingHomCompTriple σ₂' σ₂₃ σ₂₃'] [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃'] [RingHomIsometric σ₂₃'] (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) (x : E') (y : F') : ((f.bilinearComp gE gF) x) y = (f (gE x)) (gF y)

noncomputable def ContinuousLinearMap.deriv₂ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) : E × Fₗ →L[𝕜] E × Fₗ →L[𝕜] Gₗ

Derivative of a continuous bilinear map f : E →L[𝕜] F →L[𝕜] G interpreted as a map E × F → G at point p : E × F evaluated at q : E × F, as a continuous bilinear map.

Equations

• f.deriv₂ = f.bilinearComp (ContinuousLinearMap.fst 𝕜 E Fₗ) (ContinuousLinearMap.snd 𝕜 E Fₗ) + f.flip.bilinearComp (ContinuousLinearMap.snd 𝕜 E Fₗ) (ContinuousLinearMap.fst 𝕜 E Fₗ)

@[simp]

theorem ContinuousLinearMap.coe_deriv₂ {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (p : E × Fₗ) : ⇑(f.deriv₂ p) = fun (q : E × Fₗ) => (f p.1) q.2 + (f q.1) p.2

theorem ContinuousLinearMap.map_add_add {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} {Gₗ : Type u_9} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [SeminormedAddCommGroup Gₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] [NormedSpace 𝕜 Gₗ] (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x : E) (x' : E) (y : Fₗ) (y' : Fₗ) : (f (x + x')) (y + y') = (f x) y + (f.deriv₂ (x, y)) (x', y') + (f x') y'

@[simp]

theorem ContinuousLinearMap.norm_smulRight_apply {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖c.smulRight f‖ = ‖c‖ * ‖f‖

The norm of the tensor product of a scalar linear map and of an element of a normed space is the product of the norms.

@[simp]

theorem ContinuousLinearMap.nnnorm_smulRight_apply {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖c.smulRight f‖₊ = ‖c‖₊ * ‖f‖₊

The non-negative norm of the tensor product of a scalar linear map and of an element of a normed space is the product of the non-negative norms.

noncomputable def ContinuousLinearMap.smulRightL (𝕜 : Type u_1) (E : Type u_4) (Fₗ : Type u_7) [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] : (E →L[𝕜] 𝕜) →L[𝕜] Fₗ →L[𝕜] E →L[𝕜] Fₗ

ContinuousLinearMap.smulRight as a continuous trilinear map: smulRightL (c : E →L[𝕜] 𝕜) (f : F) (x : E) = c x • f.

Equations

• ContinuousLinearMap.smulRightL 𝕜 E Fₗ = { toFun := ContinuousLinearMap.smulRightₗ, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous₂ 1 ⋯

@[simp]

theorem ContinuousLinearMap.norm_smulRightL_apply {𝕜 : Type u_1} {E : Type u_4} {Fₗ : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup Fₗ] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 Fₗ] (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖((ContinuousLinearMap.smulRightL 𝕜 E Fₗ) c) f‖ = ‖c‖ * ‖f‖