Character orthogonality for compact groups (${\int }_G{\chi }_V\overline{{\chi }_W}dg=\dim {\mathrm{Hom}}_G(W,V)$)

For finite-dimensional unitary representations of a compact group, average the action on $\mathrm{Hom}(W,V)$ against normalized Haar measure. This is the orthogonal projection onto the intertwining operators, and its trace is the displayed integral. The Schur lemma therefore makes distinct irreducible characters orthogonal and each irreducible character of norm one.