Corollary

Let E and F be Banach spaces. The set $\mathscr{L}^c(E; F)$ is a closed vector subspace of the Banach space $\mathscr{L}(E; F)$. It contains the closure of the space $\mathscr{L}^f(E; F)$ of continuous linear mappings of finite rank from E into F.

This follows from Propositions 1 and 2.

There exist Banach spaces E and F such that $\mathscr{L}^f(E; F)$ is not dense in $\mathscr{L}^c(E; F) ($cf. Remark 6 of III, p. 16 and Theorem 4 of III, p. 19, b)).