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Various additions.

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Bokuan Li
2026-07-16 13:49:27 -04:00
parent 42eeae1679
commit fe557de4a6
9 changed files with 182 additions and 6 deletions
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src/fa/tvs/space-of-linear.tex
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@@ -120,11 +120,19 @@
\begin{definition}[Bounded Convergence Topology]
\label{definition:bounded-convergence-topology}
Let $E, F$ be TVSs over $K \in \RC$, $\fB \subset 2^E$ be the collection of bounded subsets of $E$, then the $\fB$-uniform topology on $L(E; F)$ is the \textbf{topology of bounded convergence}.
Let $E, F$ be TVSs over $K \in \RC$, $\fB \subset 2^E$ be the collection of bounded subsets of $E$, then the $\fB$-uniform topology on $L(E; F)$ is the \textbf{topology of bounded convergence}, or the \textbf{uniform topology}.
The space $L_b(E; F)$ denotes $L(E; F)$ equipped with the topology of bounded convergence.
\end{definition}
\begin{definition}[Topology of Precompact Convergence]
\label{definition:compact-operator-topology}
Let $E, F$ be TVSs over $K \in \RC$, $\mathfrak{K} \subset 2^E$ be the collection of precompact subsets of $E$, then the $\mathfrak{K}$-uniform topology on $L(E; F)$ is the \textbf{topology of precompact convergence}.
The space $L_c(E; F)$ denotes $L(E; F)$ equipped with the topology of precompact convergence.
\end{definition}
\begin{proposition}
\label{proposition:operator-space-completeness}
Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, then: