Proposition 9.3.4. Let $E$ be a bornologic space and $F$ be a complete Hausdorff locally convex space, then $L_{b}(E; F)$ is complete. In particular, $E^{*}$ equipped with the topology of bounded convergence is complete.
Proof. By Proposition 9.3.3, $L_{b}(E; F) = B(E; F)$. By Proposition 8.11.10, $B(E; F)$ is complete, so $L_{b}(E; F)$ is complete as well.$\square$