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Finished basic topologies on function spaces.

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Bokuan Li
2026-01-06 17:25:01 -05:00
parent 9e76c1610a
commit a704806321
5 changed files with 63 additions and 43 deletions
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src/fa/tvs/definition.tex
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@@ -130,12 +130,12 @@
\label{proposition:tvs-good-neighbourhood-base}
Let $E$ be a topological vector space over $K \in \RC$, then
\begin{enumerate}
\item $E$ admits a fundamental system of neighbourhoods at $0$ consisting of balanced and absorbing sets.
\item $E$ admits a fundamental system of neighbourhoods at $0$ consisting of circled and absorbing sets.
\item The fundamental system of neighbourhoods in $(1)$ can be taken to be open or closed.
\end{enumerate}
\end{proposition}
\begin{proof}
Firstly, (TVS2) implies that every neighbourhood of $0$ is balanced.
Firstly, (TVS2) implies that every neighbourhood of $0$ is circled.
By \ref{proposition:uniform-neighbourhoods}, $E$ admits a fundamental system of neighbourhoods consisting of open sets or closed sets.