Pontryagin Duality

Let $G$ be a locally compact Abelian Group. Let $G^*$ be the group of all homeomorphisms $G\to R/Z$, 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 $G\to (G^*)^*$, 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