Proposition 12.5.5.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 12.5.2, $L_{b}(E; F) = B(E; F)$. By Proposition 11.13.8, $B(E; F)$ is complete, so $L_{b}(E; F)$ is complete as well.$\square$
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