Added barreled spaces.

src/fa/lc/barrel.tex

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+\section{Barreled Spaces}
+\label{section:barrel}
+
+\begin{definition}[Barrel]
+\label{definition:barrel}
+    Let $E$ be a TVS over $K \in \RC$ and $D \subset E$, then $D$ is a \textbf{barrel} if it is convex, circled, radial, and closed. 
+\end{definition}
+
+\begin{definition}[Barreled Space]
+\label{definition:barreled-space}
+    Let $E$ be a locally convex space over $K \in \RC$, then the following are equivalent:
+    \begin{enumerate}
+        \item The barrels of $E$ forms a fundamental system of neighbourhoods at $0$. 
+        \item Every barrel in $E$ is a neighbourhood of $0$. 
+        \item Every lower semicontinuous seminorm on $E$ is continuous. 
+    \end{enumerate}
+\end{definition}
+\begin{proof}
+    $(2) \Rightarrow (1)$: Let $\fB \subset \cn_E(0)$ be a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets, then $\ol{\fB} = \bracsn{\ol U|U \in \fB}$ is a fundamental system of neighbourhoods at $0$ consisting of barrels. 
+
+    $(2) \Rightarrow (3)$: Let $\rho: E \to [0, \infty)$ be a lower semicontinuous seminorm, then $\bracs{\rho > 1}$ is open and $\bracs{\rho \le 1}$ is a Barrel. In which case, $\rho$ is continuous by (4) of \autoref{lemma:continuous-seminorm}. 
+
+    $(3) \Rightarrow (2)$: Let $D \subset E$ be a barrel and $\rho: E \to [0, \infty)$ be its gauge. By (4) of \autoref{definition:gauge}, $D = \bracs{\rho \le 1}$, so $\bracs{\rho > 1}$ is open, and $\rho$ is semicontinuous. By assumption, $\rho$ is continuous, so $D \in \cn_E(0)$ by (5) of \autoref{lemma:continuous-seminorm}. 
+\end{proof}
+
+\begin{summary}[Barreled Spaces]
+\label{summary:barreled-space}
+    The following types of locally convex spaces are barreled:
+    \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 Spaces of type (LB) and (LF). 
+        \item The locally convex direct sum of barreled spaces. 
+        \item Products of barreled spaces.
+    \end{enumerate}
+\end{summary}
+\begin{proof}
+    (1), (2): \autoref{proposition:baire-barrel}. 
+
+    (3), (4), (5): \autoref{proposition:barrel-limit}. 
+
+    (6): TODO. 
+\end{proof}
+
+
+\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. 
+\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, 
+    \[
+    U \subset (x + U) - (x + U) \subset nB - nB = 2nB
+    \]
+
+    so $2nB$ and thus $B$ is a neighbourhood of $0$. 
+\end{proof}
+
+\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.  
+\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. 
+\end{proof}
+