Replaced mentions of normed spaces to normed vector spaces.

src/fa/lp/definition.tex

@@ -63,7 +63,7 @@
 
 \begin{theorem}[Minkowski's Inequality, {{\cite[6.5]{Folland}}}]
 \label{theorem:minkowski}
-    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, $p \in [1, \infty]$, and $f, g \in \mathcal{L}^p(X; E)$, then
+    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $p \in [1, \infty]$, and $f, g \in \mathcal{L}^p(X; E)$, then
     \[
     \norm{f + g}_{L^p(X; E)} \le \norm{f}_{L^p(X; E)} + \norm{g}_{L^p(X; E)}
     \]
@@ -89,7 +89,7 @@
 
 \begin{definition}[$L^p$ Space]
 \label{definition:lp}
-    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, and $p \in [1, \infty]$, then $\norm{\cdot}_{L^p(X; E)}$ is a seminorm on $\mathcal{L}^p(X; E)$. The quotient
+    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $p \in [1, \infty]$, then $\norm{\cdot}_{L^p(X; E)}$ is a seminorm on $\mathcal{L}^p(X; E)$. The quotient
     \[
     L^p(X, \cm, \mu; E) = \mathcal{L}^p(X, \cm, \mu; E)/\bracs{f|f = 0\text{ a.e.}}
     \]
@@ -121,7 +121,7 @@
 
 \begin{proposition}
 \label{proposition:lp-simple-dense}
-    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, and $p \in [1, \infty)$, then $\Sigma(X, \cm; E) \cap L^p(X; E)$ is dense in $L^p(X; E)$.
+    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $p \in [1, \infty)$, then $\Sigma(X, \cm; E) \cap L^p(X; E)$ is dense in $L^p(X; E)$.
 \end{proposition}
 \begin{proof}
     Let $f \in L^p(X; E)$. By \autoref{definition:strongly-measurable}, there exists $\seq{f_n} \subset \Sigma(X, \cm; E)$ such that $\norm{f_n}_E \le \norm{f}_E$ and $\norm{f_n - f}_E \to 0$ strongly pointwise as $n \to \infty$. By the \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^p(X; E)$.
@@ -129,7 +129,7 @@
 
 \begin{theorem}[Markov's Inequality]
 \label{theorem:markov-inequality}
-    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, and $f: X \to E$ be a Borel measurable function, then
+    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $f: X \to E$ be a Borel measurable function, then
     \begin{enumerate}
         \item For any $\alpha > 0$, 
         \[
@@ -158,7 +158,7 @@
 
 \begin{proposition}
 \label{proposition:lp-in-measure}
-    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed space, $p \in [1, \infty]$, $\seq{f_n} \subset L^p(X; E)$, and $f \in L^p(X; E)$ such that $f_n \to f$ in $L^p$, then $f_n \to f$ in measure.
+    Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $p \in [1, \infty]$, $\seq{f_n} \subset L^p(X; E)$, and $f \in L^p(X; E)$ such that $f_n \to f$ in $L^p$, then $f_n \to f$ in measure.
 \end{proposition}
 \begin{proof}
     Let $\eps > 0$. If $p < \infty$, then by \hyperref[Markov's Inequality]{theorem:markov-inequality},

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 $\bracs{B(x, r)|x \in E, r > 0}$.

src/fa/rs/bv.tex

@@ -16,7 +16,7 @@
 
     is the \textbf{total variation} of $f$ on $[a, b]$ with respect to $\rho$.
 
-    If $E$ is a normed space, then the variation and total variation of $f$ is taken with respect to its norm.
+    If $E$ is a normed vector space, then the variation and total variation of $f$ is taken with respect to its norm.
 \end{definition}
 
 \begin{definition}[Bounded Variation, {{\cite[Proposition X.1.1]{Lang}}}]
@@ -35,7 +35,7 @@
         then $f \in BV([a, b]; E)$ with $[f]_{\text{var}, \rho} \le M_\rho$.
         \item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\text{var}, \rho}$.
     \end{enumerate}
-    If $(E, \norm{\cdot}_E)$ is a normed space, then
+    If $(E, \norm{\cdot}_E)$ is a normed vector space, then
     \begin{enumerate}
         \item[(5)] $f$ has at most countably many discontinuities.
     \end{enumerate}

src/measure/measurable-maps/metric.tex

@@ -125,7 +125,7 @@
 
 \begin{proposition}
 \label{proposition:measurable-simple-separable-norm}
-    Let $(X, \cm)$ be a measurable space, $(E, \norm{\cdot}_E)$ be a separable normed space, and $f: X \to E$, then the following are equivalent:
+    Let $(X, \cm)$ be a measurable space, $(E, \norm{\cdot}_E)$ be a separable normed vector space, and $f: X \to E$, then the following are equivalent:
     \begin{enumerate}
         \item $f$ is $(\cm, \cb_E)$-measurable.
         \item There exists simple functions $\seq{f_n}$ such that $\abs{f_n} \le \abs{f}$ for all $n \in \natp$, and $f_n \to f$ pointwise.

src/measure/radon/c0.tex

@@ -26,7 +26,7 @@
 
 \begin{definition}[Space of Finite Radon Measures]
 \label{definition:space-radon-measures}
-    Let $X$ be a LCH space and $E$ be a normed space over $K \in \RC$, then $M_R(X; E)$ is the \textbf{space of finite Radon measures} on $X$, which forms a vector space over $K$. 
+    Let $X$ be a LCH space and $E$ be a normed vector space over $K \in \RC$, then $M_R(X; E)$ is the \textbf{space of finite Radon measures} on $X$, which forms a vector space over $K$. 
 \end{definition}
 \begin{proof}
     Let $\mu, \nu \in M_R(X; E)$, then for any $A \in \cb_X$, $|\mu + \nu|(A) \le |\mu|(A) + |\nu|(A)$. Let $\eps > 0$, then by outer regularity and \autoref{proposition:radon-measurable-description}, there exists $K \subset A$ compact and $U \in \cn^o(A)$ such that $(|\mu| + |\nu|)(A \setminus K), (|\mu| + |\nu|)(U \setminus A) < \eps$. Therefore $|\mu + \nu|$ is regular on all Borel sets, and hence Radon. 

src/measure/radon/radon.tex

@@ -150,7 +150,7 @@
 
 \begin{proposition}
 \label{proposition:radon-cc-dense}
-    Let $X$ be a LCH space, $\mu: \cb_X \to [0, \infty]$ be a Radon measure, $E$ be a normed space, and $p \in [1, \infty)$, then $C_c(X; E)$ is dense in $L^p(X; E)$.
+    Let $X$ be a LCH space, $\mu: \cb_X \to [0, \infty]$ be a Radon measure, $E$ be a normed vector space, and $p \in [1, \infty)$, then $C_c(X; E)$ is dense in $L^p(X; E)$.
 \end{proposition}
 \begin{proof}
     By \autoref{proposition:lp-simple-dense}, $\Sigma(X, \cm; E) \cap L^p(X; E)$ is dense in $L^p(X; E)$. Using linearity, it is sufficient to approximate indicator functions of Borel sets with finite measure.

src/measure/vector/variation.tex

@@ -3,7 +3,7 @@
 
 \begin{definition}[Total Variation of Vector Measures]
 \label{definition:total-variation-vector}
-    Let $(X, \cm)$ be a measure space, $\mu$ be a vector measure taking values in a normed space $E$, and
+    Let $(X, \cm)$ be a measure space, $\mu$ be a vector measure taking values in a normed vector space $E$, and
     \[
     |\mu|(A) = \sup\bracs{\sum_{j = 1}^n \norm{\mu(A_j)}_E \bigg | \seqf{A_j} \subset \cm, A = \bigsqcup_{j = 1}^n A_j}
     \]

src/measure/vector/vector.tex

@@ -3,7 +3,7 @@
 
 \begin{definition}[Vector Measure]
 \label{definition:vector-measure}
-    Let $(X, \cm)$ be a measurable space, $E$ be a normed space over $K \in \RC$, and $\mu: \cm \to E$, then $\mu$ is a \textbf{vector measure} if:
+    Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mu: \cm \to E$, then $\mu$ is a \textbf{vector measure} if:
         \begin{enumerate}
         \item[(M1)] $\mu(\emptyset) = 0$.
         \item[(M2)] For any $\seq{A_n} \subset \cm$ pairwise disjoint, $\mu\paren{\bigsqcup_{n \in \natp}A_n} = \sum_{n \in \natp} \mu(A_n)$ where the sum converges absolutely. 
@@ -13,7 +13,7 @@
 
 \begin{proposition}
 \label{proposition:vector-measure-bounded}
-    Let $(X, \cm)$ be a measurable space, $E$ be a normed space over $K \in \RC$, and $\mu: \cm \to E$ be a vector measure, then
+    Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $\mu: \cm \to E$ be a vector measure, then
     \[
     \sup_{A \in \cm}\norm{\mu(A)}_E < \infty
     \]