Remark 2

Suppose that G is commutative. The set $\widehat{G}$ is identified with the dual group of G (V, p. 393, remark). For every $\chi \in \widehat{G}$, the space of $\boldsymbol{\varrho }_G(\chi )$ is the 1-dimensional subspace of $L^2(G)$ generated by $\chi$. Theorem 1 for G is therefore then equivalent to the corollary of theorem 1 of II, p. 215.