English
If a space of continuous multilinear maps is complete, then F is complete, provided each factor is nonzero.
Русский
Если пространство непрерывных многообразных отображений полно и каждый фактор ненулевой, то F полно.
LaTeX
$$$\text{CompleteSpace}(\mathrm{ContinuousMultilinearMap}\, \mathbb{K}\, M\, F) \land \forall i, m_i \neq 0 \Rightarrow \text{CompleteSpace}(F)$$$
Lean4
/-- A version of the **Hahn-Banach theorem**: given disjoint convex sets `s`, `t` where `s` is
closed, and `t` is compact, there is a continuous linear functional which strongly separates them.
-/
theorem geometric_hahn_banach_closed_compact (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) (ht₁ : Convex ℝ t)
(ht₂ : IsCompact t) (disj : Disjoint s t) :
∃ (f : StrongDual ℝ E) (u v : ℝ), (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b :=
let ⟨f, s, t, hs, st, ht⟩ := geometric_hahn_banach_compact_closed ht₁ ht₂ hs₁ hs₂ disj.symm
⟨-f, -t, -s, by simpa using ht, by simpa using st, by simpa using hs⟩
