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Bokuan Li
2026-01-05 20:10:39 -05:00
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src/fa/tvs/continuous.tex
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@@ -84,3 +84,41 @@
\end{enumerate}
The uniformity $\fU$ and its induced topology are the \textbf{product uniformity/topology}, and $E$ equipped with $\fU$ is the \textbf{product TVS} of $\seqi{E}$.
\end{definition}
\begin{definition}[Uniform/Bounded Operator Topology]
\label{definition:uniform-op-topo}
Let $E, F$ be TVSs over $K \in \RC$. For each bounded set $B \subset E$ and entourage $V \subset F \times F$, let
\[
U(B, V) = \bracs{(S, T) \in L(E; F) \times L(E; F)|(Sx, Tx) \in V \forall x \in B}
\]
then
\[
\fB = \bracs{U(B, V)| B \subset E \text{ bounded}, U \subset F \times F \text{ entourage}}
\]
is a fundamental system of entourages for a vector space uniformity on $L(E; F)$, which induces the \textbf{uniform/bounded operator topology} on $L(E; F)$.
\end{definition}
\begin{proof}
Firstly,
\begin{enumerate}
\item[(FB1)] Let $U(B, V), U(B', V') \in \fB$, then $U(B, V) \cap U(B', V') \supset U(B \cup B', V \cap V') \in \fB$.
\item[(UB1)] For any $U(B, V) \in \fB$, the diagonal is contained in $V$, so the diagonal is also contained in $U(B, V)$.
\item[(UB2)] For any $U(B, V) \in \fB$, there exists an entourage $W \subset F \times F$ such that $W \circ W \subset V$. So $U(B, W) \in \fB$ with
\[
U(B, W) \circ U(B, W) \subset U(B, W \circ W) \subset U(B, V)
\]
\end{enumerate}
By \ref{proposition:fundamental-entourage-criterion}, $\fB$ forms a fundamental system of entourages for a uniformity on $L(E; F)$.
TODO SCHAEFER WOLFF PAGE 79
\end{proof}
\begin{definition}[Strong Operator Topology]
\label{definition:strong-op-topo}
Let $E, F$ be TVSs over $K \in \RC$, then the \textbf{strong operator topology} on $L(E; F)$ is the initial topology generated by the maps $\bracs{T \mapsto Tx| x \in E}$, where $F$ is equipped with the strong topology.
\end{definition}
\begin{definition}[Weak Operator Topology]
\label{definition:weak-op-topo}
Let $E, F$ be TVSs over $K \in \RC$, then the \textbf{weak operator topology} on $L(E; F)$ is the initial topology generated by the maps $\bracs{T \mapsto Tx| x \in E}$, where $F$ is equipped with the weak topology.
\end{definition}