File: Mathlib/Analysis/Normed/Operator/ContinuousLinearMap.lean

full_name: LinearMap.mkContinuous
code: /-- 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`. -/
def LinearMap.mkContinuous (C : ℝ) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : E →SL[σ] F 
goal: E →SL[σ] F

full_name: LinearMap.mkContinuousOfExistsBound
code: /-- 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`. -/
def LinearMap.mkContinuousOfExistsBound (h : ∃ C, ∀ x, ‖f x‖ ≤ C * ‖x‖) : E →SL[σ] F 
goal: E →SL[σ] F

full_name: continuous_of_linear_of_boundₛₗ
code: theorem continuous_of_linear_of_boundₛₗ {f : E → F} (h_add : ∀ x y, f (x + y) = f x + f y)
    (h_smul : ∀ (c : 𝕜) (x), f (c • x) = σ c • f x) {C : ℝ} (h_bound : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    Continuous f 
goal: Continuous f

full_name: continuous_of_linear_of_bound
code: theorem continuous_of_linear_of_bound {f : E → G} (h_add : ∀ x y, f (x + y) = f x + f y)
    (h_smul : ∀ (c : 𝕜) (x), f (c • x) = c • f x) {C : ℝ} (h_bound : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    Continuous f 
goal: Continuous f

full_name: LinearMap.mkContinuous_coe
code: @[simp, norm_cast]
theorem LinearMap.mkContinuous_coe (C : ℝ) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    (f.mkContinuous C h : E →ₛₗ[σ] F) = f 
goal: ↑(f.mkContinuous C h) = f

full_name: LinearMap.mkContinuous_apply
code: @[simp]
theorem LinearMap.mkContinuous_apply (C : ℝ) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) (x : E) :
    f.mkContinuous C h x = f x 
goal: (f.mkContinuous C h) x = f x

full_name: LinearMap.mkContinuousOfExistsBound_coe
code: @[simp, norm_cast]
theorem LinearMap.mkContinuousOfExistsBound_coe (h : ∃ C, ∀ x, ‖f x‖ ≤ C * ‖x‖) :
    (f.mkContinuousOfExistsBound h : E →ₛₗ[σ] F) = f 
goal: ↑(f.mkContinuousOfExistsBound h) = f

full_name: LinearMap.mkContinuousOfExistsBound_apply
code: @[simp]
theorem LinearMap.mkContinuousOfExistsBound_apply (h : ∃ C, ∀ x, ‖f x‖ ≤ C * ‖x‖) (x : E) :
    f.mkContinuousOfExistsBound h x = f x 
goal: (f.mkContinuousOfExistsBound h) x = f x

full_name: ContinuousLinearMap.antilipschitz_of_bound
code: theorem antilipschitz_of_bound (f : E →SL[σ] F) {K : ℝ≥0} (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
    AntilipschitzWith K f 
goal: AntilipschitzWith K ⇑f

full_name: ContinuousLinearMap.bound_of_antilipschitz
code: theorem bound_of_antilipschitz (f : E →SL[σ] F) {K : ℝ≥0} (h : AntilipschitzWith K f) (x) :
    ‖x‖ ≤ K * ‖f x‖ 
goal: ‖x‖ ≤ ↑K * ‖f x‖

full_name: LinearEquiv.toContinuousLinearEquivOfBounds
code: /-- Construct a continuous linear equivalence from a linear equivalence together with
bounds in both directions. -/
def LinearEquiv.toContinuousLinearEquivOfBounds (e : E ≃ₛₗ[σ] F) (C_to C_inv : ℝ)
    (h_to : ∀ x, ‖e x‖ ≤ C_to * ‖x‖) (h_inv : ∀ x : F, ‖e.symm x‖ ≤ C_inv * ‖x‖) : E ≃SL[σ] F 
goal: E ≃SL[σ] F

full_name: LinearMap.toContinuousLinearMap₁
code: /-- 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`. -/
def LinearMap.toContinuousLinearMap₁ (f : 𝕜 →ₗ[𝕜] E) : 𝕜 →L[𝕜] E 
goal: 𝕜 →L[𝕜] E

full_name: LinearMap.toContinuousLinearMap₁_coe
code: @[simp]
theorem LinearMap.toContinuousLinearMap₁_coe (f : 𝕜 →ₗ[𝕜] E) :
    (f.toContinuousLinearMap₁ : 𝕜 →ₗ[𝕜] E) = f 
goal: ↑f.toContinuousLinearMap₁ = f

full_name: LinearMap.toContinuousLinearMap₁_apply
code: @[simp]
theorem LinearMap.toContinuousLinearMap₁_apply (f : 𝕜 →ₗ[𝕜] E) (x) :
    f.toContinuousLinearMap₁ x = f x 
goal: f.toContinuousLinearMap₁ x = f x

full_name: ContinuousLinearMap.uniformEmbedding_of_bound
code: theorem ContinuousLinearMap.uniformEmbedding_of_bound {K : ℝ≥0} (hf : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
    UniformEmbedding f 
goal: UniformEmbedding ⇑f

full_name: ContinuousLinearMap.ofHomothety
code: /-- 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.
 -/
def ContinuousLinearMap.ofHomothety (f : E →ₛₗ[σ] F) (a : ℝ) (hf : ∀ x, ‖f x‖ = a * ‖x‖) :
    E →SL[σ] F 
goal: E →SL[σ] F

full_name: ContinuousLinearEquiv.homothety_inverse
code: theorem ContinuousLinearEquiv.homothety_inverse (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
    (∀ x : E, ‖f x‖ = a * ‖x‖) → ∀ y : F, ‖f.symm y‖ = a⁻¹ * ‖y‖ 
goal: (∀ (x : E), ‖f x‖ = a * ‖x‖) → ∀ (y : F), ‖f.symm y‖ = a⁻¹ * ‖y‖

full_name: ContinuousLinearEquiv.ofHomothety
code: /-- A linear equivalence which is a homothety is a continuous linear equivalence. -/
noncomputable def ContinuousLinearEquiv.ofHomothety (f : E ≃ₛₗ[σ] F) (a : ℝ) (ha : 0 < a)
    (hf : ∀ x, ‖f x‖ = a * ‖x‖) : E ≃SL[σ] F 
goal: E ≃SL[σ] F