\section{Bornologic Spaces}
\label{section:bornologic}

\begin{definition}[Bornologic Space]
\label{definition:bornologic-space}
    Let $E$ be a locally convex space, then the following are equivalent:
    \begin{enumerate}
        \item For any $U \subset E$ convex and balanced, if $U$ absorbs every bounded set of $E$, then $U \in \cn_E(0)$.
        \item For any seminorm $\rho: E \to [0, \infty)$ that is bounded on all bounded sets of $E$, $\rho$ is continuous. 
    \end{enumerate}
    
    If the above holds, then $E$ is a \textbf{bornologic space}.
\end{definition}
\begin{proof}
    (1) $\Rightarrow$ (2): Let $B \subset E$ be bounded, then there exists $R > 0$ such that $\rho(B) \subset [0, R)$. In which case, 
    \[
    B \subset \bracs{\rho < R} = R\bracs{\rho < 1}
    \]

    By assumption, $\bracs{\rho < 1} \in \cn_E(0)$, so $\rho$ is continuous by \autoref{lemma:continuous-seminorm}.

    (2) $\Rightarrow$ (1): Let $\rho$ be the \hyperref[gauge]{definition:gauge} of $U$, then for any $B \subset E$ bounded, there exists $R > 0$ such that $B \subset RU$. In which case, $\rho(B) \subset [0, R]$. 
\end{proof}

\begin{proposition}
\label{proposition:metrisable-bornologic}
    Let $E$ be a metrisable locally convex space, then $E$ is bornologic.
\end{proposition}
\begin{proof}
    Let $U \subset E$ be convex and balanced such that $U$ absorbs every bounded set of $E$. Let $\seq{U_n} \subset \cn^o(0)$ be a decreasing countable fundamental system of neighbourhoods at $0$. If $U_n \setminus nA \ne \emptyset$ for all $n \in \natp$, then there exists $\seq{x_n}$ such that $x_n \in U_n \setminus nA$ for all $n \in \natp$. In which case, $x_n \to 0$ as $n \to \infty$, so $\seq{x_n}$ is bounded. By assumption, there exists $n \in\natp$ such that $nA \supset \seq{x_n}$, which contradicts the fact that $\seq{x_n} \cap A = \emptyset$. 
\end{proof}

\begin{proposition}
\label{proposition:bornologic-bounded}
    Let $E$ be a bornologic space, $F$ be a locally convex space, and $T \in \hom(E; F)$, then the following are equivalent:
    \begin{enumerate}
        \item $T$ is continuous.
        \item $T$ is bounded.
    \end{enumerate}
    
\end{proposition}
\begin{proof}
    (1) $\Rightarrow$ (2): By \autoref{proposition:continuous-bounded}. 

    (2) $\Rightarrow$ (1): Let $\rho: F \to [0, \infty)$ be a continuous seminorm, then $\rho \circ T$ is a seminorm on $E$ that is bounded on bounded sets. Since $E$ is bornologic, $\rho \circ T$ is continuous. Therefore $T$ is continuous by \autoref{proposition:tvs-convex-morphism}.
\end{proof}

\begin{proposition}
\label{proposition:bornologic-continuous-complete}
    Let $E$ be a bornologic space and $F$ be a complete Hausdorff locally convex space, then $L_b(E; F)$ is complete. In particular, $E^*$ equipped with the topology of bounded convergence is complete.
\end{proposition}
\begin{proof}
    By \autoref{proposition:bornologic-bounded}, $L_b(E; F) = B(E; F)$. By \autoref{proposition:operator-space-completeness}, $B(E; F)$ is complete, so $L_b(E; F)$ is complete as well.
\end{proof}