Pontryagin duality

E681628

mathematical duality theory theorem in harmonic analysis theorem in topological group theory

Pontryagin duality is a fundamental theorem in harmonic analysis and topological group theory that establishes a duality between locally compact abelian groups and their groups of continuous characters.

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Observed surface forms (1)

Surface form	Occurrences
Pontryagin duality for locally compact abelian groups	1

Statements (50)

Predicate	Object
instanceOf	mathematical duality theory ⓘ theorem in harmonic analysis ⓘ theorem in topological group theory ⓘ
appliesTo	locally compact abelian groups ⓘ
assertsIsomorphism	canonical evaluation map from a group to its double dual is an isomorphism ⓘ
codomainCategory	opposite category of locally compact abelian groups ⓘ
coreStatement	every locally compact abelian group is canonically isomorphic to its double dual ⓘ the dual of a locally compact abelian group is a locally compact abelian group ⓘ the duality functor on locally compact abelian groups is an involutive contravariant equivalence of categories ⓘ
developedBy	Lev Pontryagin NERFINISHED ⓘ
domainCategory	category of locally compact abelian groups ⓘ
failsFor	non-abelian locally compact groups in its classical form ⓘ
field	abstract harmonic analysis ⓘ harmonic analysis ⓘ topological group theory ⓘ
formalizesAs	contravariant equivalence between a category and its opposite ⓘ
hasExample	dual of a finite abelian group is isomorphic to the group itself ⓘ dual of a real vector space regarded as an additive group is isomorphic to its algebraic dual with appropriate topology ⓘ dual of the circle group is the integers ⓘ dual of the integers is the circle group ⓘ
historicalPeriod	20th century mathematics ⓘ
inspired	non-abelian duality theories ⓘ
involvesConstruction	character group ⓘ group of continuous homomorphisms into the circle group ⓘ
mathematicalArea	abstract algebra ⓘ category theory ⓘ topology ⓘ
namedAfter	Lev Pontryagin NERFINISHED ⓘ
relatedConcept	character group functor ⓘ reflexive group ⓘ
relatedResult	Gelfand duality NERFINISHED ⓘ Stone duality ⓘ
relatesConcept	Fourier analysis on groups NERFINISHED ⓘ Fourier transform NERFINISHED ⓘ continuous characters ⓘ dual group ⓘ group characters ⓘ topological groups ⓘ
requiresCondition	abelian group structure ⓘ local compactness ⓘ
specialCaseOf	duality between objects and characters ⓘ duality theory in functional analysis ⓘ
usedIn	Fourier series NERFINISHED ⓘ Fourier transform on locally compact abelian groups ⓘ Tate's thesis NERFINISHED ⓘ harmonic analysis on adelic groups ⓘ number theory ⓘ representation theory of abelian groups ⓘ
usesTargetGroup	circle group ⓘ unit complex numbers ⓘ

Referenced by (5)

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

Gelfand–Naimark theorem → isRelatedTo → Pontryagin duality ⓘ

Lev Pontryagin → notableIdea → Pontryagin duality ⓘ

this entity surface form: Pontryagin duality for locally compact abelian groups

Lev Pontryagin → notableWork → Pontryagin duality ⓘ

Gelfand transform → relatedTo → Pontryagin duality ⓘ

Gelfand representation of commutative C*-algebras → relatesTo → Pontryagin duality ⓘ