Banach–Alaoglu theorem

E424210

mathematical theorem theorem in functional analysis

The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.

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All labels observed (3)

Label	Occurrences
Banach–Alaoglu theorem canonical	2
Alaoglu’s 1940 paper on weak topologies of normed linear spaces	1
Alaoglu’s lemma	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in functional analysis ⓘ
appliesTo	dual of a locally convex space (in generalized form) ⓘ dual space of a normed space ⓘ
assumption	consider the dual space endowed with the weak-* topology ⓘ the underlying space is a normed space ⓘ
author	Leonidas Alaoglu ⓘ
characterizes	weak-* compactness of bounded sets in dual spaces ⓘ
conclusion	closed unit ball in the dual space is compact in the weak-* topology ⓘ
domain	Banach spaces ⓘ normed linear spaces ⓘ
field	functional analysis ⓘ functional analysis of Banach spaces ⓘ
generalization	extends to locally convex topological vector spaces via polars of neighborhoods of zero ⓘ
implies	closed unit ball of the dual of a Banach space is weak-* compact ⓘ every bounded net in the dual space has a weak-* convergent subnet ⓘ
importance	fundamental in modern functional analysis ⓘ key tool in proving existence of functionals and measures ⓘ
isGeneralizationOf	Alaoglu’s compactness result for duals of normed spaces ⓘ
isSpecialCaseOf	compactness of polars in the weak-* topology ⓘ
namedAfter	Leonidas Alaoglu ⓘ Stefan Banach ⓘ
oftenFormulatedAs	the unit ball of the dual of a normed space is compact in the weak-* topology ⓘ
originalPublication	Banach–Alaoglu theorem self-linksurface differs ⓘ surface form: Alaoglu’s 1940 paper on weak topologies of normed linear spaces
relatedTo	Eberlein–Šmulian theorem ⓘ Goldstine theorem ⓘ Hahn–Banach theorem ⓘ Krein–Milman theorem ⓘ Riesz representation theorem ⓘ
statement	The closed unit ball of the dual of a normed space is compact in the weak-* topology. ⓘ
topologyInvolved	weak-* topology ⓘ weak-star topology ⓘ
typeOfResult	compactness theorem ⓘ existence theorem ⓘ
usedIn	calculus of variations ⓘ distribution theory ⓘ existence proofs in functional analysis ⓘ measure theory ⓘ optimization in infinite-dimensional spaces ⓘ partial differential equations ⓘ study of dual Banach spaces ⓘ theory of Banach algebras ⓘ theory of C-algebras ⓘ weak- compactness arguments ⓘ
uses	Banach–Alaoglu theorem self-linksurface differs ⓘ surface form: Alaoglu’s lemma Tychonoff theorem for products of compact spaces ⓘ surface form: Tychonoff theorem product topology ⓘ
yearProved	1940 ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Stefan Banach → notableWork → Banach–Alaoglu theorem ⓘ

Stefan Banach → eponymOf → Banach–Alaoglu theorem ⓘ

Banach–Alaoglu theorem → uses → Banach–Alaoglu theorem self-linksurface differs ⓘ

this entity surface form: Alaoglu’s lemma

Banach–Alaoglu theorem → originalPublication → Banach–Alaoglu theorem self-linksurface differs ⓘ

this entity surface form: Alaoglu’s 1940 paper on weak topologies of normed linear spaces