Corollary

Let $u$ be a continuous linear mapping from $E_1$ into $E_2$. Then $^t u$ is continuous when the duals $E'_i$ are assigned the following topologies: a) the weak topologies $\sigma(E'_i, E_i)$; b) the strong topologies $\beta(E'_i, E_i)$; c) the Mackey topologies $\tau(E'_i, E_i)$; d) the topologies of precompact convergence. Moreover, if $E_2$ is Hausdorff, $^t u$ is continuous when the duals $E'_i$ are assigned: e) the topologies of compact convergence (resp. compact convex).

The only point which requires a proof is the case c), when the topologies of $E_1$ and $E_2$ are not necessarily Hausdorff. Then for every linear form $f \in E'_1*$, $f \circ ^t u$ is a linear form on $E'_2$; hence there is a linear mapping $v : E'_1* \to E'_2*$ which is continuous for the topologies $\sigma(E'_1*, E'_1)$ and $\sigma(E'_2*, E'_2)$ and is such that $d_{B_2} \circ u = v \circ d_{B_1}$, where $d_{B_i}$ is the canonical mapping from $E_i$ into $E'_i*$ ($i = 1, 2$). Consequently, if $A$ is a subset of $E_1$ such that $d_{B_1}(A)$ is convex, balanced and compact for $\sigma(E'_1*, E'_1)$ then $d_{B_2}(u(A)) = v(d_{B_1}(A))$ is convex, balanced and compact for $\sigma(E'_2*, E'_2)$ since the topologies $\sigma(E'_1*, E'_1)$ and $\sigma(E'_2*, E'_2)$ are Hausdorff.