Pontryagin Dual

Let be a Finite Abelian Group. The set of Characters of form a Group, called the Pontryagin dual of .

Proof

The trivial character takes everything to . It’s easy to see that if and are characters, then so is . The inverse is complex conjugate as:

Theorem

Pontryagin dual of is isomorphic to (but not in a nice way).

Proof

The isomorphism is given in the proof that Characters form an orthonormal basis for . Note that this isomorphism depends on the choice of basis for .