Corollary 4

— Let (F, G), (F_1, G_1) be two pairs in separating duality, and let u be a linear mapping of F in F_1 that is continuous for $\sigma(F, G)$ and $\sigma(F_1, G_1)$. Then u is surjective, if and only if, $^t u$ is an isomorphism of $G_1$ (with topology $\sigma(G_1, F_1)$) on $^t u(G_1)$ with the topology induced by $\sigma(G, F)$.

For, to say that $u(F) = F_1$ is equivalent to saying that $u(F)$ is closed and everywhere dense in $F_1$ for $\sigma(F_1, G_1)$; cor. 4 follows then from cor. 3 applied to $^t u$ and of II, p. 47, cor. 2.