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:

The dual group

If G is a locally compact abelian group, a character of G is a continuous group homomorphism from G with values in the circle group T. It can be shown that the set of all characters on G is itself a locally compact abelian group, called the dual group of G. The group operation on the dual group is given by pointwise multiplication of characters, the inverse of character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compact-open topology). This topology in general is not metrizable. However, if the group G is a separable locally compact abelian group, then the dual group is metrizable. The dual group of an abelian group G is denoted G^.

Related Topics:
Character - Continuous - Group homomorphism - Topology - Uniform convergence - Compact - Compact-open topology

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Theorem The dual of G^ is canonically isomorphic to G, that is

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(G^)^ = G in a canonical way.

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Canonical means that there is naturally defined map from G into (G^)^; more importantly, the map should be functorial. The precise formulation of this idea involves the concept of natural transformation. This fact is important; for instance, any finite abelian group is isomorphic to its dual, but the isomorphism is not canonical. The canonical isomorphism is defined as follows:

Related Topics:
Canonical - Functorial - Natural transformation

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: x mapsto {chi mapsto chi(x) }

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In other words, each group element x is identified to the evaluation character on the dual.

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