Theorems · Theorem · functional analysis

WeakDual.metrizable_of_isCompact

∀ (𝕜 : Type u_1) (E : Type u_3) [inst : NontriviallyNormedField 𝕜] [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [TopologicalSpace.SeparableSpace E] (K : Set (WeakDual 𝕜 E)),
  IsCompact K → TopologicalSpace.MetrizableSpace ↑K

A compact subset of the weak dual of a separable normed space is metrizable.

Defined in: Mathlib.Analysis.Normed.Module.WeakDual
Cited by: 1 results in Mathlib
Foundations: Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NontriviallyNormedField SeminormedAddCommGroup NormedSpace TopologicalSpace.SeparableSpace

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites17

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · cited by 53,352
NormedSpacestatement and proof · cited by 12,499
NontriviallyNormedFieldstatement and proof · cited by 8,742
Set.Elemstatement and proof · cited by 7,166
SeminormedAddCommGroupstatement and proof · cited by 2,671
Continuousproof · cited by 2,592
IsCompactstatement and proof · cited by 1,282
CompactSpaceproof · cited by 593
Continuous.compproof · cited by 371
Subtype.val_injectiveproof · cited by 232
continuous_subtype_valproof · cited by 159
TopologicalSpace.SeparableSpacestatement and proof · cited by 109

Cited by1

Results whose statement or proof uses this declaration.

WeakDual.isSeqCompact_of_isBounded_of_isClosedproof · cited by 2