isVonNBounded_iff — Mathlib

English

If E is complete, then E is quasi-complete for any scalar field 𝕜.

Русский

Если пространство E полноe, то оно является квази-полным для любого поля скаляров 𝕜.

LaTeX

$$$\\text{CompleteSpace}(E) \\Rightarrow \\text{QuasiCompleteSpace}_{\\mathbb{k}}(E)$$$

Lean4

theorem isVonNBounded_iff {s : Set E} : Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s :=
  by
  refine ⟨fun h ↦ ?_, isVonNBounded_of_isBounded _⟩
  rcases (h (Metric.ball_mem_nhds 0 zero_lt_one)).exists_pos with ⟨ρ, hρ, hρball⟩
  rcases NormedField.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩
  specialize hρball a ha.le
  rw [← ball_normSeminorm 𝕜 E, Seminorm.smul_ball_zero (norm_pos_iff.1 <| hρ.trans ha), ball_normSeminorm] at hρball
  exact Metric.isBounded_ball.subset hρball

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