Updated Schaefer & Wolff citations.

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Bokuan Li
2026-01-27 14:10:21 -05:00
parent 1431e56d3d
commit 2bacd9b370
5 changed files with 8 additions and 8 deletions

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@@ -43,7 +43,7 @@
\end{proof}
\begin{proposition}[{{\cite[1.1.4]{SchaeferWolff}}}]
\begin{proposition}[{{\cite[I.1.4]{SchaeferWolff}}}]
\label{proposition:tvs-uniform}
Let $E$ be a TVS over $K \in \bracs{\real, \complex}$, then:
\begin{enumerate}
@@ -85,7 +85,7 @@
\]
\end{proof}
\begin{proposition}[{{\cite[1.1.1]{SchaeferWolff}}}]
\begin{proposition}[{{\cite[I.1.1]{SchaeferWolff}}}]
\label{proposition:tvs-set-operations}
Let $E$ be a TVS over $K \in \RC$ and $A, B \subset E$, then:
\begin{enumerate}
@@ -148,7 +148,7 @@
Let $U \in \cn(0)$ be closed, then there exists a balanced neighbourhood $V \in \cn^o(0)$ such that $V \subset U$. In which case, for any $\lambda \in K$ with $0 < \abs{\lambda} \le 1$, $\lambda \overline{V} = \overline{\lambda V} \subset \overline{V}$ by (TVS2). Therefore $\overline{V} \subset U$ is balanced as well.
\end{proof}
\begin{proposition}[{{\cite[1.2]{SchaeferWolff}}}]
\begin{proposition}[{{\cite[I.1.2]{SchaeferWolff}}}]
\label{proposition:tvs-0-neighbourhood-base}
Let $E$ be a vector space over $K \in \RC$, and $\topo$ be a vector space topology on $E$, then there exists a fundamental system of neighbourhoods $\fB \subset \cn_E(0)$ such that:
\begin{enumerate}