Proposition 2

Let E be a semi-normed space and F a separated topological vector space. Suppose that every closed bounded subset of F is complete. Then the vector subspace $\mathscr{L}^c(E; F)$ of $\mathscr{L}(E; F)$ is closed for the topology of bounded convergence.

Let $u$ be an element of $\mathscr{L}(E; F)$ adhering to $\mathscr{L}^c(E; F)$ for the topology of bounded convergence. Denote by B the unit ball of E and by V a neighbourhood of 0 in F. Choose a neighbourhood W of 0 in F such that $W + W\subset V$. By hypothesis there exists an element $v$ of $\mathscr{L}^c(E; F)$ such that $(u-v)(B)\subset W$. The set $v(B)$ being relatively compact in F, there exists a finite subset M of F such that $v(B)\subset M + W$, whence $u(B)\subset M + V$. Since this holds for every V, the set $u(B)$ is precompact in F, hence its closure is also so (TG, II, prop. 1, p. 30). Since the latter is closed and bounded, it is complete by the hypothesis made on F, and consequently is compact (TG, II, cor., p. 30). It follows that $u(B)$ is relatively compact. This proves that $u$ belongs to $\mathscr{L}^c(E; F)$.

If F is locally convex, the hypothesis of the proposition means that F is quasi-complete (EVT, III, p. 8, déf. 6).