Theorems · Theorem · functional analysis

WeakDual.isCompact_closedBall

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

The Banach-Alaoglu theorem: closed balls of the dual of a normed space E are compact in the weak-star topology.

Defined in

Mathlib.Analysis.Normed.Module.WeakDual

Cited by

0 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceProperSpace