Lemma 2

Let G and H be locally compact commutative groups and let $f: H\rightarrow G$ be a morphism of topological groups. The kernel of $\widehat{f}$ is the orthogonal of the image of $f$.

By definition, we have $\chi \in$ Ker($\widehat{f}$) if and only if the restriction of $\chi$ to the image of $f$ is trivial.