Proposition 11.4.5.label Let $E$ be a bornological space and $F$ be a complete separated 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 11.4.2, $L_{b}(E; F) = B(E; F)$. By Proposition 10.12.8, $B(E; F)$ is complete, so $L_{b}(E; F)$ is complete as well.$\square$