Pontryagin Duality - The Pontryagin Duality Theorem

The Pontryagin Duality Theorem

Theorem The dual of G^ is canonically isomorphic to G, that is (G^)^ = G in a canonical way.

Canonical means that there is a naturally defined map from G into (G^)^; more importantly, the map should be functorial. The canonical isomorphism is defined as follows:

In other words, each group element x is identified to the evaluation character on the dual. This is exactly the same as the canonical isomorphism between a finite-dimensional vector space and its double dual, However, there is also a difference: V is isomorphic to its dual space V*, although not canonically so, while many groups G are not isomorphic to their dual groups (for instance, when G is T its dual is Z, and T is not isomorphic to Z as topological groups). If G is a finite abelian group, then G and G^ are isomorphic, but not canonically. To make precise the statement that there is no canonical isomorphism between finite abelian groups and their dual groups (in general) requires thinking about dualizing not only on groups, but also on maps between the groups, in order to treat dualization as a functor and prove the identity functor and the dualization functor are not naturally equivalent.