Corollary

With the notation of prop. 5, II, p. 27, suppose that $E$ is the sum of the $g_\alpha(F_\alpha)$. Let $F$ be the topological direct sum of the family $(F_\alpha)_{\alpha \in A}$, and let $j_\alpha : F_\alpha \to F$ be the canonical injection; suppose that $g : F \to E$ is the linear mapping such that $g \circ j_\alpha = g_\alpha$ for all $\alpha \in A$. If $N$ is the kernel of $g$, then the canonical bijection $F/N \to E$ associated with $g$ is a topological isomorphism of $F/N$ on $E$ with the topology $\mathscr{T}$.

This is a particular case of II, p. 28, cor. 2 remembering II, p. 29, Example I.