Added facts about vector measures.

src/fa/norm/absolute.tex

@@ -6,6 +6,22 @@
     Let $E$ be a normed space, then a series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$ \textbf{converges absolutely} if $\sum_{n \in \natp}\norm{x_n}_E < \infty$.
 \end{definition}
 
+\begin{lemma}
+\label{lemma:banach-absolute}
+    Let $E$ be a normed space, then the following are equivalent:
+    \begin{enumerate}
+        \item $E$ is a Banach space.
+        \item For any absolutely convergent series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$, there exists $x \in E$ such that $x = \sum_{n = 1}^\infty x_n$.
+    \end{enumerate}
+    
+\end{lemma}
+\begin{proof}
+    (2) $\Rightarrow$ (1): Let $\seq{x_n} \subset E$ be a Cauchy sequence, then there exists a subsequence ${n_k}$ such that $\norm{x_{n_{k+1}} - x_{n_k}} \le 2^{-k}$ for all $k \in \natp$. 
+    
+    For each $k \in \natp$, let $y_k = x_{n_{k+1}} - x_{n_k}$, then $\sum_{n = 1}^\infty y_k$ is absolutely convergent, and there exists $y \in E$ such that $y = \sum_{n = 1}^\infty y_n$. Let $x = x_{n_1} + y$, then $x_{n_k} \to x$ as $k \to \infty$. Since $\seq{x_n}$ is Cauchy, $x_n \to x$ as well by \autoref{lemma:cauchy-subnet}.
+\end{proof}
+
+
 \begin{definition}[Unconditional Convergence]
 \label{definition:unconditional-convergence}
     Let $E$ be a normed space, then a series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$ \textbf{converges unconditionally} if for any bijection $\sigma: \natp \to \natp$, 

src/fa/norm/normed.tex

@@ -31,6 +31,7 @@
     is a fundamental system of neighbourhoods at $0$ for $E$. Therefore $\norm{\cdot}_E$ induces the topology on $E$.
 \end{proof}
 
+
 \begin{theorem}[Successive Approximation]
 \label{theorem:successive-approximation}
     Let $E, F$ be normed spaces, $T \in L(E; F)$, $C \ge 0$, and $\gamma \in (0, 1)$. If for all $y \in F$, there exists $x \in E$ such that: