File: Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean

full_name: LinearMap.bound_of_shell
code: theorem bound_of_shell [RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜}
    (hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) :
    ‖f x‖ ≤ C * ‖x‖ 
goal: ‖f x‖ ≤ C * ‖x‖

full_name: LinearMap.bound_of_ball_bound
code: /-- `LinearMap.bound_of_ball_bound'` is a version of this lemma over a field satisfying `RCLike`
that produces a concrete bound.
-/
theorem bound_of_ball_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] Fₗ)
    (h : ∀ z ∈ Metric.ball (0 : E) r, ‖f z‖ ≤ c) : ∃ C, ∀ z : E, ‖f z‖ ≤ C * ‖z‖ 
goal: ∃ C, ∀ (z : E), ‖f z‖ ≤ C * ‖z‖

full_name: LinearMap.antilipschitz_of_comap_nhds_le
code: theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f 
goal: ∃ K, AntilipschitzWith K ⇑f

full_name: ContinuousLinearMap.opNorm_zero_iff
code: /-- An operator is zero iff its norm vanishes. -/
theorem opNorm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 
goal: ‖f‖ = 0 ↔ f = 0

full_name: ContinuousLinearMap.norm_id
code: /-- If a normed space is non-trivial, then the norm of the identity equals `1`. -/
@[simp]
theorem norm_id [Nontrivial E] : ‖id 𝕜 E‖ = 1 
goal: ‖ContinuousLinearMap.id 𝕜 E‖ = 1

full_name: ContinuousLinearMap.nnnorm_id
code: @[simp]
lemma nnnorm_id [Nontrivial E] : ‖id 𝕜 E‖₊ = 1 
goal: ‖ContinuousLinearMap.id 𝕜 E‖₊ = 1

full_name: ContinuousLinearMap.normOneClass
code: instance normOneClass [Nontrivial E] : NormOneClass (E →L[𝕜] E) 
goal: NormOneClass (E →L[𝕜] E)

full_name: ContinuousLinearMap.toNormedAddCommGroup
code: /-- Continuous linear maps themselves form a normed space with respect to
    the operator norm. -/
instance toNormedAddCommGroup [RingHomIsometric σ₁₂] : NormedAddCommGroup (E →SL[σ₁₂] F) 
goal: NormedAddCommGroup (E →SL[σ₁₂] F)

full_name: ContinuousLinearMap.toNormedRing
code: /-- Continuous linear maps form a normed ring with respect to the operator norm. -/
instance toNormedRing : NormedRing (E →L[𝕜] E) 
goal: NormedRing (E →L[𝕜] E)

full_name: ContinuousLinearMap.homothety_norm
code: theorem homothety_norm [RingHomIsometric σ₁₂] [Nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ}
    (hf : ∀ x, ‖f x‖ = a * ‖x‖) : ‖f‖ = a 
goal: ‖f‖ = a

full_name: ContinuousLinearMap.antilipschitz_of_embedding
code: /-- If a continuous linear map is a topology embedding, then it is expands the distances
by a positive factor. -/
theorem antilipschitz_of_embedding (f : E →L[𝕜] Fₗ) (hf : Embedding f) :
    ∃ K, AntilipschitzWith K f 
goal: ∃ K, AntilipschitzWith K ⇑f

full_name: LinearIsometry.norm_toContinuousLinearMap
code: @[simp]
theorem norm_toContinuousLinearMap [Nontrivial E] [RingHomIsometric σ₁₂] (f : E →ₛₗᵢ[σ₁₂] F) :
    ‖f.toContinuousLinearMap‖ = 1 
goal: ‖f.toContinuousLinearMap‖ = 1

full_name: LinearIsometry.norm_toContinuousLinearMap_comp
code: /-- Postcomposition of a continuous linear map with a linear isometry preserves
the operator norm. -/
theorem norm_toContinuousLinearMap_comp [RingHomIsometric σ₁₂] (f : F →ₛₗᵢ[σ₂₃] G)
    {g : E →SL[σ₁₂] F} : ‖f.toContinuousLinearMap.comp g‖ = ‖g‖ 
goal: ‖f.toContinuousLinearMap.comp g‖ = ‖g‖

full_name: ContinuousLinearMap.opNorm_comp_linearIsometryEquiv
code: /-- Precomposition with a linear isometry preserves the operator norm. -/
theorem opNorm_comp_linearIsometryEquiv (f : F →SL[σ₂₃] G) (g : F' ≃ₛₗᵢ[σ₂'] F) :
    ‖f.comp g.toLinearIsometry.toContinuousLinearMap‖ = ‖f‖ 
goal: ‖f.comp g.toLinearIsometry.toContinuousLinearMap‖ = ‖f‖

full_name: ContinuousLinearMap.norm_smulRight_apply
code: /-- 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 norm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖ = ‖c‖ * ‖f‖ 
goal: ‖c.smulRight f‖ = ‖c‖ * ‖f‖

full_name: ContinuousLinearMap.nnnorm_smulRight_apply
code: /-- 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. -/
@[simp]
theorem nnnorm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖₊ = ‖c‖₊ * ‖f‖₊ 
goal: ‖c.smulRight f‖₊ = ‖c‖₊ * ‖f‖₊

full_name: ContinuousLinearMap.smulRightL
code: /-- `ContinuousLinearMap.smulRight` as a continuous trilinear map:
`smulRightL (c : E →L[𝕜] 𝕜) (f : F) (x : E) = c x • f`. -/
def smulRightL : (E →L[𝕜] 𝕜) →L[𝕜] Fₗ →L[𝕜] E →L[𝕜] Fₗ 
goal: (E →L[𝕜] 𝕜) →L[𝕜] Fₗ →L[𝕜] E →L[𝕜] Fₗ

full_name: ContinuousLinearMap.norm_smulRightL_apply
code: @[simp]
theorem norm_smulRightL_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRightL 𝕜 E Fₗ c f‖ = ‖c‖ * ‖f‖ 
goal: ‖((ContinuousLinearMap.smulRightL 𝕜 E Fₗ) c) f‖ = ‖c‖ * ‖f‖

full_name: ContinuousLinearMap.norm_smulRightL
code: @[simp]
theorem norm_smulRightL (c : E →L[𝕜] 𝕜) [Nontrivial Fₗ] : ‖smulRightL 𝕜 E Fₗ c‖ = ‖c‖ 
goal: ‖(ContinuousLinearMap.smulRightL 𝕜 E Fₗ) c‖ = ‖c‖

full_name: ContinuousLinearMap.norm_smulRightL_le
code: lemma norm_smulRightL_le : ‖smulRightL 𝕜 E Fₗ‖ ≤ 1 
goal: ‖ContinuousLinearMap.smulRightL 𝕜 E Fₗ‖ ≤ 1

full_name: Submodule.norm_subtypeL
code: theorem norm_subtypeL (K : Submodule 𝕜 E) [Nontrivial K] : ‖K.subtypeL‖ = 1 
goal: ‖K.subtypeL‖ = 1

full_name: ContinuousLinearEquiv.antilipschitz
code: protected theorem antilipschitz (e : E ≃SL[σ₁₂] F) :
    AntilipschitzWith ‖(e.symm : F →SL[σ₂₁] E)‖₊ e 
goal: AntilipschitzWith ‖↑e.symm‖₊ ⇑e

full_name: ContinuousLinearEquiv.one_le_norm_mul_norm_symm
code: theorem one_le_norm_mul_norm_symm [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    1 ≤ ‖(e : E →SL[σ₁₂] F)‖ * ‖(e.symm : F →SL[σ₂₁] E)‖ 
goal: 1 ≤ ‖↑e‖ * ‖↑e.symm‖

full_name: ContinuousLinearEquiv.norm_pos
code: theorem norm_pos [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    0 < ‖(e : E →SL[σ₁₂] F)‖ 
goal: 0 < ‖↑e‖

full_name: ContinuousLinearEquiv.norm_symm_pos
code: theorem norm_symm_pos [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    0 < ‖(e.symm : F →SL[σ₂₁] E)‖ 
goal: 0 < ‖↑e.symm‖

full_name: ContinuousLinearEquiv.nnnorm_symm_pos
code: theorem nnnorm_symm_pos [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
    0 < ‖(e.symm : F →SL[σ₂₁] E)‖₊ 
goal: 0 < ‖↑e.symm‖₊

full_name: ContinuousLinearEquiv.subsingleton_or_norm_symm_pos
code: theorem subsingleton_or_norm_symm_pos [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
    Subsingleton E ∨ 0 < ‖(e.symm : F →SL[σ₂₁] E)‖ 
goal: Subsingleton E ∨ 0 < ‖↑e.symm‖

full_name: ContinuousLinearEquiv.subsingleton_or_nnnorm_symm_pos
code: theorem subsingleton_or_nnnorm_symm_pos [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
    Subsingleton E ∨ 0 < ‖(e.symm : F →SL[σ₂₁] E)‖₊ 
goal: Subsingleton E ∨ 0 < ‖↑e.symm‖₊

full_name: ContinuousLinearEquiv.coord_norm
code: @[simp]
theorem coord_norm (x : E) (h : x ≠ 0) : ‖coord 𝕜 x h‖ = ‖x‖⁻¹ 
goal: ‖ContinuousLinearEquiv.coord 𝕜 x h‖ = ‖x‖⁻¹

full_name: IsCoercive
code: /-- A bounded bilinear form `B` in a real normed space is *coercive*
if there is some positive constant C such that `C * ‖u‖ * ‖u‖ ≤ B u u`.
-/
def IsCoercive [NormedAddCommGroup E] [NormedSpace ℝ E] (B : E →L[ℝ] E →L[ℝ] ℝ) : Prop 
goal: Prop

full_name: NormedSpace.equicontinuous_TFAE
code: /-- Equivalent characterizations for equicontinuity of a family of continuous linear maps
between normed spaces. See also `WithSeminorms.equicontinuous_TFAE` for similar characterizations
between spaces satisfying `WithSeminorms`. -/
protected theorem NormedSpace.equicontinuous_TFAE : List.TFAE
    [ EquicontinuousAt ((↑) ∘ f) 0,
      Equicontinuous ((↑) ∘ f),
      UniformEquicontinuous ((↑) ∘ f),
      ∃ C, ∀ i x, ‖f i x‖ ≤ C * ‖x‖,
      ∃ C ≥ 0, ∀ i x, ‖f i x‖ ≤ C * ‖x‖,
      ∃ C, ∀ i, ‖f i‖ ≤ C,
      ∃ C ≥ 0, ∀ i, ‖f i‖ ≤ C,
      BddAbove (Set.range (‖f ·‖)),
      (⨆ i, (‖f i‖₊ : ENNReal)) < ⊤ ] 
goal: [EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),
    ∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖, ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖,
    ∃ C, ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (Set.range fun x => ‖f x‖),
    ⨆ i, ↑‖f i‖₊ < ⊤].TFAE