Pontryagin Duality

Let G be a locally compact Abelian group. Let G^* be the group of all continuous homeomorphisms G->R/Z, in the compact open topology. Then G^* is also a locally compact Abelian group, where the asterisk defines a contravariant equivalence of the category of locally compact Abelian groups with itself. The natural mapping G->(G^*)^*, sending g to G, where G(f)=f(g), is an isomorphism and a homeomorphism. Under this equivalence, compact groups are sent to discrete groups and vice versa.

See also

Abelian Group, Homeomorphism

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Cite this as:

Weisstein, Eric W. "Pontryagin Duality." From MathWorld--A Wolfram Web Resource.

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