Remark

— 1) Let $(F_1, G_1), (F_2, G_2), (F_3, G_3)$ be three pairs of spaces in separating duality and consider a sequence of two linear mappings

that are continuous for the weak topologies corresponding respectively with $G_1, G_2, G_3$; we consider the sequence of transposed mappings

It is clear that $^t(v \circ u) = ^t u \circ ^t v$, therefore the relation $v \circ u = 0$ is equivalent to $^t u \circ ^t v = 0$. The sequence (5) is exact if, and only if, the three following conditions are satisfied

a) $^t u \circ ^t v = 0$; b) $\operatorname{Im}(^t v)$ is dense in $\operatorname{Ker}(^t u)$; c) $^t u$ is a strict morphism of $G_2$ in $G_1$.

This follows in effect from cor. 3 of II, p. 49 and formulae (3) and (4) of II, p. 47.

2) It must not be thought that when $u$ is a strict morphism of $F$ in $F_1$, then $^t u$ is necessarily a strict morphism of $G_1$ in $G$; in other words $u$ can be a strict morphism without $u(F)$ being closed in $F_1$ for $\sigma(F_1, G_1)$. This is shown by the example where $F$ is a non-closed subspace of $F_1$ and $G = G_1 / F^\circ$, $u$ being the canonical injection. Similarly, the fact that the sequence (5) is exact does not necessarily imply that (6) is exact, however, if the sequence (5) is exact and if $v$ is a strict morphism, then the sequence (6) is exact, by the remark 1 and by II, p. 49, cor. 3.