Corollary 3

— Let (F, G), (F_1, G_1) be two pairs of spaces in separating duality and let u be a linear mapping of F in F_1, which is continuous for $\sigma(F, G)$ and $\sigma(F_1, G_1)$. Then u is a strict morphism of F in F_1, if and only if, Im($^t u$) is a closed subspace in G for $\sigma(G, F)$.

Let $N = \operatorname{Im}(^t u) \subset G$; we know that $N^0 = \operatorname{Ker}(u)$ in F (II, p. 47, formula (3)). Let $p : F \to F/N^\circ$ be the canonical mapping so that u factorises as

where w is injective. The spaces $F/N^\circ$ and N are in separating duality and by formula (1) of II, p. 48, we have $\langle w(\dot{y}), z_1 \rangle = \langle \dot{y}, ^t u(z_1) \rangle$ for all $\dot{y} \in F/N^\circ$ and $z_1 \in G_1$. This relation shows that w is an isomorphism of $F/N^\circ$, carrying the topology $\sigma(F/N^\circ, N)$, on $u(F)$ with the topology induced by $\sigma(F_1, G_1)$. The conclusion results therefore from cor. 1 and the definition of a strict morphism.