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Updated spelling of barreled to barrelled for consistency.

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Bokuan Li
2026-05-06 16:50:37 -04:00
parent 7e6e37d3e8
commit 57c32a3c5e
2 changed files with 18 additions and 9 deletions
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14
src/fa/lc/barrel.tex
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@@ -25,14 +25,14 @@
\begin{summary}
\label{summary:barreled-space}
The following types of locally convex spaces are barreled:
The following types of locally convex spaces are barrelled:
\begin{enumerate}
\item Every locally convex space with the Baire property.
\item Every Banach space and every Fréchet space.
\item Inductive limits of barreled spaces.
\item Inductive limits of barrelled spaces.
\item Spaces of type (LB) and (LF).
\item The locally convex direct sum of barreled spaces.
\item Products of barreled spaces.
\item The locally convex direct sum of barrelled spaces.
\item Products of barrelled spaces.
\end{enumerate}
\end{summary}
\begin{proof}
@@ -46,7 +46,7 @@
\begin{proposition}
\label{proposition:baire-barrel}
Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barreled.
Let $E$ be a locally convex space over $K \in \RC$. If $E$ is a Baire space, then $E$ is barrelled.
\end{proposition}
\begin{proof}[Proof, {{\cite[II.7.1]{SchaeferWolff}}}. ]
Let $D \subset E$ be a Barrel, then $E = \bigcup_{n \in \natp}nD$ is a countable union of closed sets. Since $E$ is Baire, there exists $n \in \natp$, $U \in \cn_E(0)$ circled, and $x \in E$ such that $x + U \in nB$. In which case,
@@ -59,9 +59,9 @@
\begin{proposition}
\label{proposition:barrel-limit}
Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barreled.
Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barrelled.
\end{proposition}
\begin{proof}[Proof, {{\cite[II.7.2]{SchaeferWolff}}}. ]
Let $D \subset E$ be a barrel, then for each $i \in I$, $T_i^{-1}(D) \subset E_i$ is also a barrel, and thus a neighbourhood of $0$ in $E_i$. By (5) of \autoref{definition:lc-inductive}, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barreled.
Let $D \subset E$ be a barrel, then for each $i \in I$, $T_i^{-1}(D) \subset E_i$ is also a barrel, and thus a neighbourhood of $0$ in $E_i$. By (5) of \autoref{definition:lc-inductive}, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barrelled.
\end{proof}