Theorems · Theorem · functional analysis

geometric_hahn_banach_open

∀ {E : Type u_2} [inst : TopologicalSpace E] [inst_1 : AddCommGroup E] [inst_2 : Module ℝ E] {s t : Set E}
  [IsTopologicalAddGroup E] [ContinuousSMul ℝ E],
  Convex ℝ s → IsOpen s → Convex ℝ t → Disjoint s t → ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b

A version of the Hahn-Banach theorem: given disjoint convex sets s, t where s is open, there is a continuous linear functional which separates them.

Defined in

Mathlib.Analysis.LocallyConvex.Separation

Cited by

4 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceAddCommGroupModuleIsTopologicalAddGroupContinuousSMul