Corollary

Suppose that the duality between F and G is separating in G. If u is a linear mapping of F in $F_1$, that is continuous for $\sigma(F, G)$ and $\sigma(F_1, G_1)$, then its transpose is a linear mapping of $G_1$ in $G$, that is continuous for $\sigma(G_1, F_1)$ and $\sigma(G, F)$. Further if the duality between $F_1$ and $G_1$ is separating in $F_1$ then ${}^t({}^t u) = u$.

It is sufficient to exchange $F$ and $F_1$ with $G$ and $G_1$ in prop. 5.