Theorems · Theorem · functional analysis

RCLike.geometric_hahn_banach_of_nonempty_interior_point

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : TopologicalSpace E] [inst_1 : AddCommGroup E] [inst_2 : Module ℝ E] {x : E}
  [inst_3 : RCLike 𝕜] [inst_4 : Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E]
  {A : Set E},
  Convex ℝ A → x ∉ interior A → (interior A).Nonempty → ∃ f, f ≠ 0 ∧ ∀ a ∈ A, RCLike.re (f a) ≤ RCLike.re (f x)

If A is convex with nonempty interior and x ∉ interior A, then there is a nonzero continuous 𝕜-linear functional whose real part attains its maximum on A at x.

Defined in

Mathlib.Analysis.LocallyConvex.Separation

Cited by

0 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceAddCommGroupModuleRCLikeModuleIsScalarTowerIsTopologicalAddGroupContinuousSMul