Added more bits on bornologic spaces.

src/fa/tvs/bounded.tex

@@ -3,8 +3,19 @@
 
 \begin{definition}[Bounded]
 \label{definition:bounded}
-    Let $E$ be a TVS over $K \in \RC$ and $B \subset E$, then $B$ is \textbf{bounded} if for every $U \in \cn(0)$, there exists $\lambda \in K$ such that $\lambda U \supset B$. The collection $B(E)$ is the set of all bounded sets of $E$.
+    Let $E$ be a TVS over $K \in \RC$ and $B \subset E$, then the following are equivalent:
+    \begin{enumerate}
+        \item For every $U \in \cn(0)$, there exists $\lambda \in K$ such that $\lambda U \supset B$. 
+        \item For every $\seq{x_n} \subset B$ and $\seq{\lambda_n} \subset K$ such that $\lambda_n \to 0$, $\lambda_n x_n \to 0$ as $n \to \infty$.
+    \end{enumerate}
+
+    If the above holds, then $B$ is \textbf{bounded}. The collection $B(E)$ is the set of all bounded sets of $E$.
 \end{definition}
+\begin{proof}
+    (1) $\Rightarrow$ (2): Let $U \in \cn_E(0)$ be circled, then there exists $k \in \natp$ such that $kU \supset B$. Since $\lambda_n \to 0$ as $n \to \infty$, there exists $N \in \natp$ such that $|\lambda_n| \le 1/k$ for all $n \ge N$. In which case, $\lambda_n x_n \in \lambda_n B \subset U$ for all $n \ge N$. 
+
+    $\neg (1) \Rightarrow \neg (2)$: Let $U \in \cn_E(0)$ be circled such that $B \not\subset nU$ for all $n \in \natp$. For each $n \in \natp$, let $x_n \in B \setminus nU$, then $x_n/n \not\in U$ for all $n \in \natp$. 
+\end{proof}
 
 \begin{proposition}[{{\cite[I.5.1]{SchaeferWolff}}}]
 \label{proposition:bounded-operations}