dual group of $G$ is homeomorphic to the character space of $L^1(G)$

Let $G$ be a locally compact abelian (http://planetmath.org/AbelianGroup2) group (http://planetmath.org/TopologicalGroup) and $L^1(G)$ its group algebra.

Let $\widehat{G}$ denote the Pontryagin dual of $G$ and $\mathrm{\Delta }$ the character space of $L^1(G)$, i.e. the set of multiplicative linear functionals of $L^1(G)$ endowed with the weak-* topology.

Theorem - The spaces $\widehat{G}$ and $\mathrm{\Delta }$ are homeomorphic. The homeomorphism is given by

$$\omega ⟼{\varphi }_{\omega },\omega \in \widehat{G}$$

where ${\varphi }_{\omega }\in \mathrm{\Delta }$ is defined by

$${\varphi }_{\omega }(f):={\int }_{G}f(s)\omega (s)𝑑\mu (s),f\in L^1(G)$$