Exercise 4, § 3

a) Let $E_1, E_2, F$ be three topological vector spaces (over $\mathbf{R}$ or $\mathbf{C}$). Let $f$ be a continuous linear mapping from $E_1$ into $E_2$, and $\mathfrak{S}_1$ (resp. $\mathfrak{S}_2$) a family of bounded subsets of $E_1$ (resp. $E_2$), such that $f(\mathfrak{S}_1) \subset \mathfrak{S}_2$. Show that $u \mapsto u \circ f$ is a continuous linear mapping from $\mathscr{L}_{\mathfrak{S}_2}(E_2 ; F)$ into $\mathscr{L}_{\mathfrak{S}_1}(E_1 ; F)$. b) Let $E, F$ be two topological vector spaces, and $M$ be a vector subspace of $E$. Let $f$ be the canonical map from $E$ onto $E/M$, and $\mathfrak{S}$ be a family of bounded subsets of $E$. Show that the mapping $u \mapsto u \circ f$ is an isomorphism from $\mathscr{L}_{f(\mathfrak{S})}(E/M ; F)$ onto the subspace of $\mathscr{L}_{\mathfrak{S}}(E ; F)$ consisting of those continuous linear mappings from $E$ into $F$ which are null on $M$.