geometric_hahn_banach_closed_point

English

If the space of continuous multilinear maps is complete, then F is complete, under suitable nontriviality.

Русский

Полнота пространства непрерывных многообразных отображений в F сохраняется при ненулевых условиях.

LaTeX

$$$\text{CompleteSpace}(\mathrm{ContinuousMultilinearMap}\, 𝕜\, M\, F) \Rightarrow \text{CompleteSpace}(F)$$$

Lean4

theorem geometric_hahn_banach_closed_point (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) (disj : x ∉ s) :
    ∃ (f : StrongDual ℝ E) (u : ℝ), (∀ a ∈ s, f a < u) ∧ u < f x :=
  let ⟨f, s, _t, ha, hst, hb⟩ :=
    geometric_hahn_banach_closed_compact hs₁ hs₂ (convex_singleton x) isCompact_singleton
      (disjoint_singleton_right.2 disj)
  ⟨f, s, ha, hst.trans <| hb x <| mem_singleton _⟩

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