## locally compact abelian

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• Further generalisation to locally compact abelian groups is required in number theory.
• The real line is a locally compact abelian group.
• A locally compact abelian group G is compact if and only if the dual group G^ is discrete.
• However, if the group G is a separable locally compact abelian group, then the dual group is metrizable.
• The study of locally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelian locally compact groups.
• There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.
• The representation theory for locally compact abelian groups is described by Pontryagin duality.
• It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.
• In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms.
• The locally compact abelian case is part of the Pontryagin duality theory.
• The following books have chapters on locally compact abelian groups, duality and Fourier transform.
• See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.
• For locally compact abelian groups, the character group (with an assumption of continuity) is central to Fourier analysis.
• In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters.
• The Hausdorff-Young inequality can also be established for the Fourier transform on locally compact abelian groups.
• The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform.
• One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context.
• He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan.
• More precisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups.
• As a locally compact abelian group, the adeles have a nontrivial translation invariant measure.