Proposition 3

— Let E be a locally convex metrizable space, $E'_b$ its strong dual and G a Fréchet space. The space $\mathscr{L}_b(E'_b ; G)$ is a Fréchet space.

By prop. 2 (IV, p. 22), there exists a sequence $(A_n)$ of bounded subsets of $E'_b$ such that every bounded subset of $E'_b$ is contained in one of the $A_n$. Let $(V_n)$ be a countable fundamental system of neighbourhoods of 0 in G. Let $H_{mn}$ be the set of linear mappings $u$ from $E'_b$ into G such that $u(A_m) \subset V_n$. Then $(H_{mn})$ is a fundamental system of neighbourhoods of 0 in $\mathscr{L}_b(E'_b ; G)$, and the latter space is then metrizable.

To show that $\mathscr{L}_b(E'_b ; G)$ is complete, it is enough to prove that every Cauchy sequence $(u_n)$ in this space is convergent; since G is complete, there exists a linear mapping $u : E'_b \to G$ such that $(u_n)$ converges simply to u. By IV, p. 23, corollary, we have $u \in \mathscr{L}_b(E'_b ; G)$. It then follows from prop. 5 of GT, X, § 1, No. 5, that $(u_n)$ converges to u in $\mathscr{L}_b(E'_b ; G)$.