\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}
\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}

