Exercise 12, § 1

Let $(G_i)_{i\in I}$ be a family of locally compact commutative groups. For every $i\in I$, let $H_i$ be an open compact subgroup of $G_i$. Let G be the direct local product of the $G_i$ relative to the $H_i$ (TG, III, p. 71, exerc. 26); it is locally compact. For every $\chi \in \widehat{G}$, let $\chi_i$ be the restriction of $\chi$ to $G_i$. Now prove that $\chi \mapsto (\chi_i)_{i\in I}$ is an isomorphism of the topological group $\widehat{G}$ onto the direct local product of the $\widehat{G}_i$ relative to the $H^{\bot}_i$.