Theorems · Theorem · functional analysis

geometric_hahn_banach_of_nonempty_interior_point

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

If A is convex with nonempty interior and x ∉ interior A, then there is a nonzero continuous linear functional whose maximum on A is attained at x.

Defined in

Mathlib.Analysis.LocallyConvex.Separation

Cited by

1 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceAddCommGroupModuleIsTopologicalAddGroupContinuousSMul