Theorems · Theorem · functional analysis

ContinuousLinearMap.extend_eq

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [inst : AddCommGroup E]
  [inst_1 : UniformSpace E] [inst_2 : IsUniformAddGroup E] [inst_3 : AddCommGroup F] [inst_4 : UniformSpace F]
  [inst_5 : IsUniformAddGroup F] [inst_6 : T0Space F] [inst_7 : AddCommMonoid Eₗ] [inst_8 : UniformSpace Eₗ]
  [inst_9 : ContinuousAdd Eₗ] [inst_10 : Semiring 𝕜] [inst_11 : Semiring 𝕜₂] [inst_12 : Module 𝕜 E]
  [inst_13 : Module 𝕜₂ F] [inst_14 : Module 𝕜 Eₗ] [inst_15 : ContinuousConstSMul 𝕜 Eₗ]
  [inst_16 : ContinuousConstSMul 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} (f : E →SL[σ₁₂] F) [inst_17 : CompleteSpace F] {e : E →L[𝕜] Eₗ},
  DenseRange ⇑e → IsUniformInducing ⇑e → ∀ (x : E), (f.extend e) (e x) = f x

Defined in

Mathlib.Analysis.Normed.Operator.Extend

Cited by

5 results in Mathlib

Foundations

Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupUniformSpaceIsUniformAddGroupAddCommGroupUniformSpaceIsUniformAddGroupT0SpaceAddCommMonoidUniformSpaceContinuousAddSemiringSemiringModuleModuleModuleContinuousConstSMulContinuousConstSMulCompleteSpace