Added the support function.
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Bokuan Li
2026-06-27 13:17:45 -04:00
parent eed3a342e4
commit 968fbe6eba
5 changed files with 141 additions and 12 deletions

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@@ -1,7 +1,7 @@
\section{Equicontinuous Families of Linear Maps}
\label{section:equicontinuous-linear}
\begin{proposition}[{{\cite[IV.4.2]{SchaeferWolff}}}]
\begin{proposition}
\label{proposition:equicontinuous-linear}
Let $E, F$ be TVSs over $K \in RC$ and $\alg \subset \hom(E; F)$, then the following are equivalent:
\begin{enumerate}
@@ -12,7 +12,7 @@
\item For each $V \in \cn_F(0)$, $\bigcap_{T \in \alg}T^{-1}(V) \in \cn_E(0)$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{proof}[Proof, {{\cite[IV.4.2]{SchaeferWolff}}}. ]
(5) $\Rightarrow$ (1): Let $V \in \cn_F(0)$, then $U = \bigcap_{T \in \alg}T^{-1}(V) \in \cn_E(0)$. Thus for any $x, y \in E$ with $x - y \in U$, $Tx - Ty \in V$ for all $T \in \alg$.
\end{proof}