Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

Bohr compactification and almost-periodicity

One important application of Pontryagin duality is the following characterization of compact abelian topological groups:

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Theorem. A locally compact abelian group G is compact iff the dual group G^ is discrete. Conversely,

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G is discrete iff G^ is compact.

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The Bohr compactification is defined for any topological group G, regardless of whether G is locally compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete abelian groups is to characterize the Bohr compactification of an arbitrary abelian locally compact topological group. The Bohr compactification B(G) of G is H^, where H has the group structure G^, but given the discrete topology. Since the inclusion map

Related Topics:
Bohr compactification - Discrete topology - Inclusion map

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: iota: H ightarrow widehat{G}

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is continuous and a homomorphism, the dual morphism

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: G sim widehat{widehat{G}} { ightarrow} widehat{H}

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is a morphism into a compact group which is easily shown to satisfy the requisite universal property.

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See also almost periodic function.

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