Corollary 2

Let $G, G'$ be two metrizable abelian topological groups; let $u : G \to G'$ be a strict morphism with kernel $N$ and image $P$. Then $\hat{u} : \hat{G} \to \hat{G}'$ is a strict morphism with kernel $\hat{N}$ and image $\hat{P}$.

Let $u = j \circ v \circ p$ be the canonical factorization of $u$, where $v$ is an isomorphism of the topological group $G/N$ onto the topological group $u(G) = P$. We have $\hat{u} = \hat{j} \circ \hat{v} \circ \hat{p}$, and we have seen that $\hat{p}$ is the canonical mapping of $\hat{G}$ onto $\hat{G}\hat{N}$, and that $\hat{j}$ is the canonical mapping of $\hat{P}$ into $\hat{G}$. On the other hand, $\hat{v}$ is an isomorphism of $\hat{G}/\hat{N}$ onto $\hat{P}$ (Chapter III, § 3, no. 4, Proposition 5), whence the result.