Exercise 11, § 2

Let F, G be two Hausdorff locally convex spaces, G being assumed quasi-complete; let $u$ be a continuous linear mapping of F into G; its transpose $t_u$ being strongly continuous, the transpose $t(t_u)$ is a strongly and weakly continuous linear mapping of $F''$ into $G''$. Show that the following conditions are equivalent:

$\alpha)$ $u$ transforms every bounded subset of F into a relatively compact subset of G for $\sigma(G, G')$.

$\beta)$ $t(t_u)$ maps $F''$ into G.

$\gamma)$ $t_u$ is continuous for the topologies $\sigma(G', G)$ and $\sigma(F', F'')$.

$\delta)$ $t_u$ transforms every equicontinuous subset of $G'$ into a set relatively compact for $\sigma(F', F'')$.

(Use the fact that every bounded subset of F is relatively compact in $F''$ for $\sigma(F'', F')$. To see that $\delta$ implies $\gamma$), consider first the case that G is complete, and use Cor. 1 of Th. 2 of TVS, III, §3, No. 6. To pass to the general case, embed G in its completion and observe that every point of $F''$ is in the closure, for the topology $\sigma(F'', F')$, of a bounded subset of F.)