Theorems · Theorem · functional analysis

Module.Dual.exists_continuous_extension_of_le_seminorm

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : AddCommGroup E] [inst_1 : NormedField 𝕜] [IsRCLikeNormedField 𝕜]
  [inst_3 : TopologicalSpace E] [inst_4 : Module 𝕜 E] [PolynormableSpace 𝕜 E] (S : Submodule 𝕜 E) (f : Module.Dual 𝕜 ↥S)
  {p : Seminorm 𝕜 E},
  Continuous ⇑p → (∀ (x : ↥S), ‖f x‖ ≤ p ↑x) → ∃ g, (∀ (x : ↥S), g ↑x = f x) ∧ ∀ (x : E), ‖g x‖ ≤ p x

Hahn-Banach theorem for linear functionals dominated by a continuous seminorm on polynormable spaces over fields satisfying IsRCLikeNormedField.

Defined in

Mathlib.Analysis.LocallyConvex.HahnBanach

Cited by

2 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupNormedFieldIsRCLikeNormedFieldTopologicalSpaceModulePolynormableSpace