Corollary 2

— Suppose that E and F are metrizable and that E is barrelled. Then $\phi$ is a topological vector space isomorphism from $\mathscr{B}(E, F)$ onto $\mathscr{L}_b(E; F'_b)$.

Let $f \in \mathscr{L}_b(E; F'_b)$. Put $u(x, y) = \langle y, f(x) \rangle$ for $x \in E$ and $y \in F$. The bilinear form $u$ on $E \times F$ is separately continuous; by prop. 2, it belongs to $\mathscr{B}(E, F)$, and we have $f = \phi(u)$. Hence $\phi$ is a linear bijection from $\mathscr{B}(E, F)$ onto $\mathscr{L}_b(E; F'_b)$. It is immediate that $\phi$ is bicontinuous, hence cor. 2 follows.