Added facts about vector measures.

src/fa/lc/bornologic.tex

@@ -0,0 +1,56 @@
+\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}
+
+

src/fa/lc/convex.tex

@@ -99,10 +99,16 @@
         \item $[\cdot]$ is uniformly continuous.
         \item $[\cdot]$ is continuous.
         \item $[\cdot]$ is continuous at $0$.
+        \item $\bracs{x \in E| [x] < 1} \in \cn_E(0)$.
     \end{enumerate}
 \end{lemma}
 \begin{proof}
-    $(3) \Rightarrow (1)$: Let $\eps > 0$, then there exists $V \in \cn(0)$ such that $[x] < \eps$ for all $x \in V$. In which case, for any $x, y \in E$ with $x - y \in V$, $\abs{[x] - [y]} \le [x - y] < \eps$. By \autoref{proposition:tvs-uniform}, $[\cdot]$ is uniformly continuous.
+    $(4) \Rightarrow (1)$: Let $x, y \in E$ and $r > 0$. If
+    \[
+    x - y \in \bracs{x \in E|[x] < r} = r\bracs{x \in E|[x] < 1} \in \cn_E(0)
+    \]
+    
+    then $[x - y] < r$.
 \end{proof}
 
 

src/fa/lc/index.tex

@@ -4,6 +4,7 @@
 
 \input{./convex.tex}
 \input{./continuous.tex}
+\input{./bornologic.tex}
 \input{./quotient.tex}
 \input{./projective.tex}
 \input{./inductive.tex}