Proposition 6

Let $u$ be a linear mapping from $E_1$ into $E_2$ which is continuous for the weakened topologies. For $i = 1, 2$, let $\mathfrak{S}_i$ be a family of bounded subsets of $E_i$. In order that $^t u$ is a continuous mapping from $(E'_2)_{\mathfrak{S}_2}$ into $(E'_1)_{\mathfrak{S}_1}$, it is necessary and sufficient that, for every set $A \in \mathfrak{S}_1$, there exist sets $A_1, ..., A_n$ in $\mathfrak{S}_2$ and a real number $\lambda > 0$ such that $\lambda \cdot u(A)$ is contained in the closed convex balanced envelope of $A_1 \cup ... \cup A_n$\footnote{In other words, $u(\mathfrak{S}_1)$ is contained in the smallest adapted bornology containing $\mathfrak{S}_2$ (III, p. 3).}.

This is an immediate consequence of prop. 2 of III, p. 15.