Theorems · Theorem · functional analysis

LinearMap.extendOfNorm_eq

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [inst : NormedDivisionRing 𝕜]
  [inst_1 : NormedDivisionRing 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_2 : AddCommGroup E] [inst_3 : SeminormedAddCommGroup Eₗ]
  [inst_4 : NormedAddCommGroup F] [inst_5 : Module 𝕜 E] [inst_6 : Module 𝕜₂ F] [inst_7 : IsBoundedSMul 𝕜₂ F]
  [inst_8 : Module 𝕜 Eₗ] [inst_9 : IsBoundedSMul 𝕜 Eₗ] [inst_10 : CompleteSpace F] {f : E →ₛₗ[σ₁₂] F} {e : E →ₗ[𝕜] Eₗ},
  DenseRange ⇑e → (∃ C, ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖) → ∀ (x : E), (f.extendOfNorm e) (e x) = f x

Defined in

Mathlib.Analysis.Normed.Operator.Extend

Cited by

7 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedDivisionRingNormedDivisionRingAddCommGroupSeminormedAddCommGroupNormedAddCommGroupModuleModuleIsBoundedSMulModuleIsBoundedSMulCompleteSpace