vonNBornology_eq — Mathlib

English

The von Neumann bornology equals the metric bornology on a normed space.

Русский

Фон Неймана борология равна метрической борологии в нормированном пространстве.

LaTeX

$$$\\operatorname{vonNBornology}_{\\mathbb{k}}(E) = \\mathsf{toBornology}(\\text{Metric})$$$

Lean4

/-- In a normed space, the von Neumann bornology (`Bornology.vonNBornology`) is equal to the
metric bornology. -/
theorem vonNBornology_eq : Bornology.vonNBornology 𝕜 E = PseudoMetricSpace.toBornology :=
  by
  rw [Bornology.ext_iff_isBounded]
  intro s
  rw [Bornology.isBounded_iff_isVonNBounded]
  exact isVonNBounded_iff _

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