Theorems · Theorem · functional analysis

WeakDual.isSeqCompact_closedBall

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

The Sequential Banach-Alaoglu theorem: closed balls of the dual of a separable normed space V are sequentially compact in the weak-\* topology.

Defined in

Mathlib.Analysis.Normed.Module.WeakDual

Cited by

0 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceTopologicalSpace.SeparableSpaceProperSpace