General definition for a locally compact abelian group
Suppose is a locally compact abelian group. The Pontryagin dual of , denoted , is defined as follows:
- As an abstract group, it is the group of all continuous homomorphisms from to the circle group, with pointwise multiplication (these homomorphisms are called characters, though that term has other related meanings too).
- The topology on the set is that of uniform convergence on compact sets.
Definition for a finite group
Note that treating as a discrete topological group, this definition agrees with the previous definition, and also comes with the discrete topology.
- The Pontryagin dual of a locally compact abelian group is also a locally compact abelian group. Thus, the operation of taking Pontryagin duals can be iterated.
- Pontryagin duality theorem: The canonical homomorphism from a locally compact abelian group to its double Pontryagin dual (i.e., the dual of its dual) is an isomorphism. Thus, being Pontryagin dual is a symmetric relationship.
- Finite abelian group is isomorphic to its Pontryagin dual: Note, however, that this isomorphism is not canonical.
- The Pontryagin dual of a compact abelian group is discrete, and the Pontryagin dual of a discrete abelian group is compact. In particular, in the infinite case, it is not necessarily true that a group be isomorphic to its Pontryagin dual.