Replaced mentions of normed spaces to normed vector spaces.

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
     \[
     \sup_{A \in \cm}\norm{\mu(A)}_E < \infty
     \]