Theorems · Theorem · functional analysis

WeakDual.isClosed_closedBall

∀ {𝕜 : Type u_1} {E : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] (x' : StrongDual 𝕜 E) (r : ℝ), IsClosed (⇑WeakDual.toStrongDual ⁻¹' Metric.closedBall x' r)

Closed balls in StrongDual 𝕜 E pull back to closed sets in WeakDual 𝕜 E.

Defined in

Mathlib.Analysis.Normed.Module.WeakDual

Cited by

3 results in Mathlib

Foundations

Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpace