Corollary 3

Let $\pi_1$ and $\pi_2$ be irreducible representations of G. In the Hilbert space $L^2(G)$, one has $\langle \chi_{\pi_1}|\chi_{\pi_2}\rangle = 1$ if $\pi_1$ and $\pi_2$ are isomorphic and $\langle \chi_{\pi_1}|\chi_{\pi_2}\rangle = 0$ otherwise. In other words, the family of the characters of the classes $\pi \in \widehat{G}$ is an orthonormal family in $L^2(G)$.

This results from Proposition 1 of V, p. 457 and Corollary 2, noting that for every orthonormal basis $(e_i)_{i\in I}$ of the space of a finite-dimensional unitary representation $\pi$ of G, and for every $g\in G$, one has the formula