isClosed_closedBall — Mathlib

English

For any x' ∈ StrongDual 𝕜 E and radius r, the preimage of the closed ball under the weak-strong dual identification is closed.

Русский

Для любого x' ∈ StrongDual 𝕜 E и радиуса r об образ предобраза замкнутого шара под идентификацией слабой и сильной двойственности замкнуто.

LaTeX

$$$IsClosed(toStrongDual^{-1}'(\\text{closedBall}(x', r)))$$$

Lean4

theorem isClosed_closedBall (x' : StrongDual 𝕜 E) (r : ℝ) : IsClosed (toStrongDual ⁻¹' closedBall x' r) :=
  isClosed_induced_iff'.2 (ContinuousLinearMap.is_weak_closed_closedBall x' r)

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