Lemma 5

The space $A(\widehat{G})$ is dense in $L^2(\widehat{G})$.

Let $h$ be an element of $L^2(\widehat{G})$ orthogonal to $\widehat{A}(G)$. For $f$ and $g$ in A(G), one has $\mathscr{F}(f)\cdot \mathscr{F}(g) =\mathscr{F}(f*g)\in \widehat{A}(G)$, hence $h\cdot \overline{\mathscr{F}(f)}$ is orthogonal to $\mathscr{F}(g)$. Thus, for every $f\in A(G)$, the function $h\cdot \mathscr{F}(f)$ is orthogonal to $\widehat{A}(G)$. But $h\cdot \mathscr{F}(f)\in L^1(\widehat{G})$, and $\widehat{A}(G)$ is dense in $\mathscr{C}_0(\widehat{G})$, therefore the measure $h\mathscr{F}(f)\cdot \nu$ is zero, that is to say that $h\mathscr{F}(f)$ is $\nu$-locally negligible (INT, V, §5, n$^o3$, cor. 2 of prop. 3). In particular, $h$ is $\nu$-locally negligible on the set $\Omega_f$ of characters $\chi$ such that $\mathscr{F}(f)(\chi )\not= 0$. By Corollary 2, it follows that $h$ is $\nu$-locally negligible, hence zero since $h$ belongs to $L^2(\widehat{G})$. This concludes the proof.