Pontryagin dual

Let $G$ be a locally compact abelian group and $\mathbb{T}$ the 1-torus, i.e. the unit circle in $\mathbb{C}$ .

Definition - A continuous homomorphism $G \longrightarrow \mathbb{T}$ is called a character of $G$ . The set of all characters is called the Pontryagin dual of $G$ and is denoted by $\hat{G}$ .

Under pointwise multiplication $\hat{G}$ is also an abelian group. Since $\hat{G}$ is a group of functions we can make it a topological group under the compact-open topology (topology of convergence on compact sets).

Examples

• $\hat{\mathbb{Z}} \cong \mathbb{T}$ , via $n \mapsto z^n$ with $z \in \mathbb{T}$ .
• $\hat{\mathbb{T}} \cong \mathbb{Z}$ , via $z \mapsto z^n$ with $n \in \mathbb{Z}$ .
• $\hat{\mathbb{R}} \cong \mathbb{R}$ , via $t \mapsto e^{ist}$ with $s \in \mathbb{R}$ .

Properties

The following are some important properties of the dual group:

Theorem - Let $G$ be a locally compact abelian group. We have that

• $\hat{G}$ is also locally compact.
• $\hat{G}$ is second countable if and only if $G$ is second countable.
• $\hat{G}$ is compact if and only if $G$ is discrete.
• $\hat{G}$ is discrete if and only if $G$ is compact.
• $\widehat{(\oplus_{i \in J} G_i)} \cong \oplus_{i \in J} \hat{G_i}$ for any finite set $J$ . This isomorphism is natural.

Pontryagin duality

Let $f:G \longrightarrow H$ be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map $\hat{f}:\hat{H} \longrightarrow \hat{G}$ defined by

Theorem - The dualization $\hat{\;}:\mathord{\mathbf{LcA}} \longrightarrow \mathord{\mathbf{LcA}} $ is a contravariant functor from the category of locally compact abelian groups to itself.

Isomorphism with the second dual

Although in general there is not a canonical identification of $G$ with its dual $\hat{G}$ , there is a natural isomorphism between $G$ and its dual's dual $\hat{\hat{G}}$ :

Theorem - The map $G \longrightarrow \hat{\hat{G}}$ defined by $s \mapsto \hat{\hat{s}}$ , where $\hat{\hat{s}}(\phi) := \phi(s)$ , is a natural isomorphism between $G$ and $\hat{\hat{G}}$ .

Applications

The study of dual groups allows one to visualize Fourier series, Fourier transforms and discrete Fourier transforms from a more abstract and unified view-point, providing the basis for a general definition of Fourier transform. Thus, dual groups and Pontryagin duality are the foundations of the theory of abstract abelian harmonic analysis.