Replaced mentions of normed spaces to normed vector spaces.

src/fa/norm/absolute.tex

@@ -3,12 +3,12 @@
 
 \begin{definition}[Absolute Convergence]
 \label{definition:absolute-convergence}
-    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 $\sum_{n \in \natp}\norm{x_n}_E < \infty$.
+    Let $E$ be a normed vector 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:
+    Let $E$ be a normed vector 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 $x = \sum_{n = 1}^\infty x_n$.
@@ -24,7 +24,7 @@
 
 \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 $\sigma: \natp \to \natp$, 
+    Let $E$ be a normed vector space, then a series $\sum_{n = 1}^\infty x_n$ with $\seq{x_n} \subset E$ \textbf{converges unconditionally} if for any bijection $\sigma: \natp \to \natp$, 
     \[
     \sum_{n = 1}^\infty x_n = \sum_{n = 1}^\infty x_{\sigma(n)}
     \]

src/fa/norm/index.tex

@@ -1,4 +1,4 @@
-\chapter{Normed Spaces}
+\chapter{Normed Vector Spaces}
 \label{chap:normed-spaces}
 
 \input{./normed.tex}

src/fa/norm/multilinear.tex

@@ -3,7 +3,7 @@
 
 \begin{proposition}
 \label{proposition:bilinear-separate}
-    Let $E, F, G$ be normed spaces and $T: E \times F \to G$ be a bilinear map. If:
+    Let $E, F, G$ be normed vector spaces and $T: E \times F \to G$ be a bilinear map. If:
     \begin{enumerate}
         \item For each $x \in E$, $y \mapsto T(x, y)$ is a continuous linear map from $F$ to $G$.
         \item For each $y \in F$, $x \mapsto T(x, y)$ is a continuous linear map from $E$ to $G$.

src/fa/norm/normed.tex

@@ -34,7 +34,7 @@
 
 \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:
+    Let $E, F$ be normed vector 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:
     \begin{enumerate}
         \item[(a)] $\norm{x}_E \le C\norm{y}_F$.
         \item[(b)] $\norm{y - Tx}_F \le \gamma \norm{y}_F$.
@@ -69,7 +69,7 @@
 
 \begin{theorem}[Uniform Boundedness Principle]
 \label{theorem:uniform-boundedness}
-    Let $E, F$ be normed spaces and $\mathcal{T} \subset L(E; F)$. If
+    Let $E, F$ be normed vector spaces and $\mathcal{T} \subset L(E; F)$. If
     \begin{enumerate}
         \item For every $x \in E$, $\sup_{T \in \mathcal{T}}\norm{Tx}_F < \infty$.
         \item $E$ is a Banach space.
@@ -90,7 +90,7 @@
 
 \begin{proposition}
 \label{proposition:dual-norm}
-    Let $E$ be a normed space, then for any $x \in E$, 
+    Let $E$ be a normed vector space, then for any $x \in E$, 
     \[
     \norm{x}_E = \sup_{\substack{\phi \in E^* \\ \norm{\phi}_{E^*} = 1}}\dpn{x, \phi}{E}
     \]

src/fa/norm/separable.tex

@@ -1,9 +1,9 @@
-\section{Separable Normed Spaces}
+\section{Separable Normed Vector Spaces}
 \label{section:separable-banach-space}
 
 \begin{proposition}
 \label{proposition:separable-dual}
-    Let $E$ be a separable normed space, then $E^*$ is separable with respect to the weak*-topology. 
+    Let $E$ be a separable normed vector space, then $E^*$ is separable with respect to the weak*-topology. 
 \end{proposition}
 \begin{proof}
     Let $\seq{x_n} \subset E$ be a dense subset and $S = \bracsn{\phi \in E^*| \norm{\phi}_{E^*} \le 1}$. For each $N \in \natp$, let
@@ -23,7 +23,7 @@
 
 \begin{proposition}
 \label{proposition:separable-banach-borel-sigma-algebra}
-    Let $E$ be a separable normed space, then the Borel $\sigma$-algebra on $E$ is generated by the following families of sets:
+    Let $E$ be a separable normed vector space, then the Borel $\sigma$-algebra on $E$ is generated by the following families of sets:
     \begin{enumerate}
         \item Open sets in $E$ with respect to the strong topology. 
         \item $\bracs{B(x, r)|x \in E, r > 0}$.