Theorems · Theorem · functional analysis

WeakDual.isSeqCompact_of_isBounded_of_isClosed

∀ (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] {s : Set (WeakDual 𝕜 E)},
  Bornology.IsBounded s → IsClosed s → IsSeqCompact s

Bounded closed sets in the weak dual of a separable normed space are sequentially compact.

Defined in

Mathlib.Analysis.Normed.Module.WeakDual

Cited by

2 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceTopologicalSpace.SeparableSpaceProperSpace