Corollary

Suppose that G is compact and that $dx$ is the normalized Haar measure on G. Then the family of unitary characters of G is an orthonormal basis of $L^2(G)$.

Since G is compact, the characters of G belong to $L^2(G)$. For $\chi$ and $\xi$ in $\widehat{G}$, we have

therefore the family of characters of G is orthonormal (prop. 6 of II, p. 210). It is moreover total because the scalar product of $\chi \in \widehat{G}$ and of $f\in L^2(G)$ is equal to $\mathscr{F}_G(f)(\chi )$, and therefore $f$ is orthogonal to $\widehat{G}$ if and only if $\mathscr{F}_G(f)$ is zero in $L^2(\widehat{G})$, if and only if $f$ is zero (th. 1).