Corollary 3

Let $G, G', G''$ be three metrizable abelian topological groups and let $u : G \to G'$ and $v : G' \to G''$ be two strict morphisms such that the sequence $G \xrightarrow{u} G' \xrightarrow{v} G''$ is exact [i.e., $u(G) = \overline{v}(0)$]. Then the sequence $\hat{G} \xrightarrow{\hat{u}} \hat{G}' \xrightarrow{\hat{v}} \hat{G}''$ is exact.

For if we put $N = u(G) = \overline{v}(0)$, it follows from Corollary 2 that $\hat{N}$ is both the image of $\hat{u}$ and the kernel of $\hat{v}$.