# Pontryagin dual

## Definition

### General definition for a locally compact abelian group

Suppose $G$ is a locally compact abelian group. The Pontryagin dual of $G$, denoted $\hat{G}$, is defined as follows:

• As an abstract group, it is the group of all continuous homomorphisms from $G$ 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

Suppose $G$ is a finite abelian group. The Pontryagin dual of $G$, denoted $\hat{G}$ is the finite group of all homomorphisms from $G$ to the circle group.

Note that treating $G$ as a discrete topological group, this definition agrees with the previous definition, and $\hat{G}$ also comes with the discrete topology.

## Facts

• 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.