Theorems · Theorem · functional analysis

LinearMap.opNorm_extendOfNorm_le

∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_3} {Eₗ : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_2 : NormedAddCommGroup F]
  [inst_3 : SeminormedAddCommGroup Eₗ] [inst_4 : NormedSpace 𝕜₂ F] [inst_5 : NormedSpace 𝕜 Eₗ] [inst_6 : AddCommGroup E]
  [inst_7 : Module 𝕜 E] [inst_8 : CompleteSpace F] {f : E →ₛₗ[σ₁₂] F} {e : E →ₗ[𝕜] Eₗ},
  DenseRange ⇑e → ∀ {C : ℝ}, 0 ≤ C → (∀ (x : E), ‖f x‖ ≤ C * ‖e x‖) → ‖f.extendOfNorm e‖ ≤ C

Defined in

Mathlib.Analysis.Normed.Operator.Extend

Cited by

0 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAddCommGroupSeminormedAddCommGroupNormedSpaceNormedSpaceAddCommGroupModuleCompleteSpace