Remark

— Let E be as in th. 2; let F be a Hausdorff locally convex space which is the union of the images of a sequence of continuous linear mappings $u_n : F_n \to F$, where each $F_n$ is a Fréchet space satisfying the first axiom of countability; then $\mathscr{L}_c(E ; F)$ is a Lusin space. As in prop. 1, we first reduce to the case where each $u_n$ is injective; then, as in the proof of th. 2, we can assume that E is a Fréchet space satisfying the first axiom of countability. Then, by I, p. 20, prop. 1, $\mathscr{L}(E ; F)$ is the union of the $\mathscr{L}(E ; F_n)$; moreover, the canonical injection $\mathscr{L}_c(E ; F_n) \to \mathscr{L}_c(E ; F)$ is continuous (GT, X, § 1, No. 4, prop. 3). Since each of the spaces $\mathscr{L}_c(E ; F_n)$ is a Lusin space by th. 2, $\mathscr{L}(E ; F_n)$ is also a Lusin space for the topology induced by that of $\mathscr{L}_c(E ; F)$ (GT, IX, § 6, No. 4, prop. 11); consequently $\mathscr{L}_c(E ; F)$ is a Lusin space by virtue of GT, IX, § 6, No. 7, corollary of th. 3.