Definition 33.1.1 (Unitary Character).label Let $G$ be a locally compact Abelian group and $\phi: G \to \mathbb{T}$, then $\phi$ is a unitary character of $G$ if $\phi$ is a continuous homomorphism.

The set $\wh G$ of all unitary characters of $G$, equipped with pointwise operations and the topology of compact convergence, is the dual group of $G$.

For any $\phi \in \wh G$ and $x \in G$, denote $\phi(x) = \dpn{x, \phi}{G}$.