Remark 1

Suppose that G is a commutative group. By corollary 7 of V, p. 390, every irreducible unitary representation of G in a complex Hilbert space is of dimension 1. If G is locally compact, the set $\widehat{G}$ is identified with the set of unitary characters of G (def. 1 of II, p. 201), and the notation $\widehat{G}$ is therefore compatible with that introduced in def. 2 of II, p. 201.