Plancherel theorem for locally compact abelian groups ($\Vert \mathcal{F}f{\Vert }_{L^2(\hat{G})}=\Vert f{\Vert }_{L^2(G)}$)

For $h\in C_c(G)$, apply Fourier inversion on a locally compact abelian group at the identity to $h\ast h^{\ast }$, whose transform is $|\hat{h}{|}^2$. 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.