If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$.
Show that the linear span of $\widehat{G}$ is dense in $C(G)$,the continuous function space on $G$.
I try to solve this using stone-weierstrass theorem. But I need to prove that $\forall 0\neq g_0\in G$,there is $\gamma_0 \in \widehat{G}$ such that $\gamma_0(g_0)\neq 1$.
Any help would be appreciated!