Corollary 5

Let $\varphi : G\rightarrow H$ be a morphism of locally compact commutative groups. Then the subgroup Im($\varphi$ ) of H and the subgroup Ker($\widehat{\varphi}$) of $\widehat{H}$ are orthogonal to one another. In particular, in order that $\widehat{\varphi}$ be injective, it is necessary and sufficient that the image of $\varphi$ be dense in H.

We have Ker($\widehat{\varphi}$) $=\varphi (G)^{\bot}$ (Lemma 2 of II, p. 205), whence the result by Corollary 3.