= Plancherel theorem for locally compact abelian groups
{c}
{title2=$\|\mathcal Ff\|_{L^2(\widehat G)}=\|f\|_{L^2(G)}$}
For $h\in C_c(G)$, apply <Fourier inversion on a locally compact abelian group> at the identity to $h*h^*$, whose transform is $|\widehat h|^2$. This proves the displayed norm identity on $C_c(G)$. Density and completeness give a unique linear isometric extension to all $L^2(G)$. On $L^1\cap L^2$ it agrees almost everywhere with the integral <Fourier transform on a locally compact abelian group>.
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