Pontryagin duality
E681628
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Pontryagin duality canonical | 9 |
| Pontryagin duality for locally compact abelian groups | 1 |
| Pontryagin duality theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7685045 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Pontryagin duality Context triple: [Lev Pontryagin, notableWork, Pontryagin duality]
-
A.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
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B.
Gelfand transform
The Gelfand transform is a fundamental construction in functional analysis that represents elements of a commutative Banach algebra as continuous functions on its space of maximal ideals, linking algebraic structure with topological and spectral properties.
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C.
Alexander duality
Alexander duality is a theorem in algebraic topology that relates the homology (or cohomology) of a subspace of a sphere to the reduced cohomology of its complement.
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D.
Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
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E.
Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Pontryagin duality Target entity description: 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.
-
A.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
-
B.
Gelfand transform
The Gelfand transform is a fundamental construction in functional analysis that represents elements of a commutative Banach algebra as continuous functions on its space of maximal ideals, linking algebraic structure with topological and spectral properties.
-
C.
Alexander duality
Alexander duality is a theorem in algebraic topology that relates the homology (or cohomology) of a subspace of a sphere to the reduced cohomology of its complement.
-
D.
Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
-
E.
Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
- F. None of above. chosen
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 ⓘ |
| 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 ⓘ |
| relatedConcept |
character group functor
ⓘ
reflexive group ⓘ |
| relatedResult |
Gelfand duality
ⓘ
Stone duality ⓘ |
| relatesConcept |
Fourier analysis on groups
ⓘ
Fourier transform ⓘ 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
ⓘ
Fourier transform on locally compact abelian groups ⓘ Tate's thesis ⓘ harmonic analysis on adelic groups ⓘ number theory ⓘ representation theory of abelian groups ⓘ |
| usesTargetGroup |
circle group
ⓘ
unit complex numbers ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Pontryagin duality Description of subject: 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.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Pontryagin duality
linked to: Pontryagin duality