Exercise 7, § 3

Let $E, F_1, F_2$ be three topological vector spaces, let $f$ be a continuous linear mapping from $F_1$ into $F_2$, and $\mathfrak{S}$ be a set of bounded subsets of $E$; show that $u \mapsto f \circ u$ is a continuous linear mapping from $\mathscr{L}_{\mathfrak{S}}(E; F_1)$ into $\mathscr{L}_{\mathfrak{S}}(E; F_2)$.