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Used "separated" instead of Hausdorff in the context of topological vector spaces.

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Bokuan Li
2026-05-01 13:32:08 -04:00
parent caf7790b15
commit 3077563278
6 changed files with 8 additions and 8 deletions
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src/fa/lc/bornologic.tex
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@@ -47,7 +47,7 @@
\begin{proposition}
\label{proposition:bornologic-continuous-complete}
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.
Let $E$ be a bornologic 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.
\end{proposition}
\begin{proof}
By \autoref{proposition:bornologic-bounded}, $L_b(E; F) = B(E; F)$. By \autoref{proposition:operator-space-completeness}, $B(E; F)$ is complete, so $L_b(E; F)$ is complete as well.