Corollary

With the same hypotheses on E and F as in prop. 10 and assuming that E is Hausdorff, then every continuous surjective mapping v of F in E is a strict morphism.

Let N be the kernel of $v$ and write $N_n = j_n^{-1}(N)$; then the mapping $j'_n : F_n / N_n \to F / N$, deduced from $j_n$ by taking quotients, is injective and continuous, also $F_n / N_n$ is a Fréchet space (since $N_n$ is closed) and $F / N$ is the union of the images under $j'_n$. By hypothesis, in the canonical factorisation $v : F \to F / N \xrightarrow{w} E$, the linear mapping $w$ is bijective and continuous and its graph in $(F / N) \times E$ is therefore closed (GT, I, § 8.1, cor. 2 of prop. 2). By the remarks at the beginning and by prop. 10, the inverse mapping $u$ of $w$ is therefore continuous and the corollary is proved.

\* Prop. 10 and its corollary apply in particular when E is a complete bornological space (III, p. 12) and F is the inductive limit of a sequence of Fréchet spaces. \*