Theorems · Theorem · functional analysis

RCLike.geometric_hahn_banach_closed_point

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : TopologicalSpace E] [inst_1 : AddCommGroup E] [inst_2 : Module ℝ E] {s : Set E}
  {x : E} [inst_3 : RCLike 𝕜] [inst_4 : Module 𝕜 E] [IsScalarTower ℝ 𝕜 E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E]
  [LocallyConvexSpace ℝ E],
  Convex ℝ s → IsClosed s → x ∉ s → ∃ f u, (∀ a ∈ s, RCLike.re (f a) < u) ∧ u < RCLike.re (f x)

Defined in

Mathlib.Analysis.LocallyConvex.Separation

Cited by

3 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceAddCommGroupModuleRCLikeModuleIsScalarTowerIsTopologicalAddGroupContinuousSMulLocallyConvexSpace