I wanted to prove that for an abelian group $G$ , $\phi : G \rightarrow \hat{\hat{G}}$ is an isomorphism where $\hat{G}$ is a set of all irreducible characters of $G$
for $x \in G$, $[\phi(x)](\chi) = \chi(x)$.
My approach : I have already proved that $\vert \hat{G}\vert = \vert G \vert $ Now I thought its enough to say that for any $g \in G$ there exists an irreducible character $\chi$ such that $\chi(g) \neq 1 \forall g \in G$ and thus this map is injection. I dont know what next ..
If $\chi_{reg}$ is the character of the regular representation, then we have $\chi_{reg}(g) = 0$ for all nontrivial $g \in G$. On the other hand, it is a positive integral combination of irreducible characters (in fact it is just the sum of all of them). In particular we can't have a positive integral combination of positive real numbers summing to 0, so for all nontrivial $g$ we can't have $\chi(g) = 1$ for all irreducible characters.