Corollary 2

Let G be a locally compact commutative group. The group G is finite if and only if $\widehat{G}$ is finite. A closed subgroup H of G is of finite index if and only if its orthogonal is finite.

By duality, the first assertion follows from the fact that the dual of a finite group is finite (corollary 1). The second results from it, since $\widehat{G}/H$ is identified with $H^{\bot}$ (th. 4 of II, p. 226).