Proposition 11

Let E and F be locally convex spaces. Suppose that E is an approximation space.

a) The set $\mathscr{L}^f(E; F)$ is dense in $\mathscr{L}_{pc}(E; F)$;

b) The set $\mathscr{L}^f(F; E)$ is dense in $\mathscr{L}_{pc}(F; E)$;

c) Let $\mathfrak{S}$ be a set of bounded subsets of F and $u\in \mathscr{L}(F; E)$. Suppose that the image under $u$ of every subset of F belonging to $\mathfrak{S}$ is precompact. Then $u$ is adherent to $\mathscr{L}^f(F; E)$ for the $\mathfrak{S}$-topology (EVT, III, p. 13).

Let $v$ be an element of $\mathscr{L}(E; F)$. The mapping $\varphi :w\mapsto v\circ w$ of $\mathscr{L}_{pc}(E)$ into $\mathscr{L}_{pc}(E; F)$ is continuous (TG, X, p. 5, prop. 3). We have $\varphi (1_E) =v$ and $\varphi (\mathscr{L}^f(E))\subset \mathscr{L}^f(E; F)$, therefore $v$ is adherent to $\mathscr{L}^f(E; F)$ in $\mathscr{L}_{pc}(E; F)$. This proves a).

Analogously, under the assumptions of c), the mapping $\psi :w\mapsto w\circ u$ of $\mathscr{L}_{pc}(E)$ into $\mathscr{L}_{\mathfrak{S}}(F; E)$ is continuous (loc. cit.). We have $\psi (1_E) =u$ and $\psi (\mathscr{L}^f(E))\subset \mathscr{L}^f(F; E)$, therefore $u$ is adherent to $\mathscr{L}^f(F; E)$ in $\mathscr{L}_{\mathfrak{S}}(F; E)$.

The image of a precompact subset of F by a continuous linear mapping of F into E is precompact, therefore b) follows from c).