Corollary

Let E be a separated approximation space and F a locally convex space. Every compact linear mapping of F into E is adherent to $\mathscr{L}^f(F; E)$ in the space $\mathscr{L}(F; E)$ endowed with the topology of bounded convergence.

This follows from prop. 11, c), for the image of a bounded subset of F by a compact linear mapping of F into E is relatively compact in E (remark 1 of III, p. 2), therefore precompact.