Remark

— It is clear that the mappings $u \in \mathscr{R}(L^1, F')$ are continuous for every $\mathscr{G}$-topology on $F'$ ($\mathscr{G}$ a covering of $F$ by bounded subsets). Conversely, if $F$ is assumed moreover to be barreled, then every continuous linear mapping of $L^1$ into $F'$ equipped with an $\mathscr{G}$-topology transforms the unit ball of $L^1$ into a bounded subset of $F'$, which is therefore equicontinuous (TVS, III, §4, No. 2, Th. 1).