Added the Mackey-Arens theorem.

src/dg/complex/runge.tex

@@ -16,7 +16,7 @@
     \end{enumerate}
 \end{proposition}
 \begin{proof}[Proof, {{\cite[Proposition VIII.1.1]{ConwayComplex}}}. ]
-    Via \hyperref[fattening]{proposition:distance-compact}, let $V \in \cn_\complex^o(K)$ precompact with $\ol V \subset U$. Identify $\complex = \real^2$, then since $V$ is precompact, there exists $\delta > 0$ and $\seqf{(x_j, y_j)} \subset U$ such that:
+    Via \hyperref[fattening]{proposition:distance-compact}, let $V \in \cn_\complex^o(K)$ relatively compact with $\ol V \subset U$. Identify $\complex = \real^2$, then since $V$ is relatively compact, there exists $\delta > 0$ and $\seqf{(x_j, y_j)} \subset U$ such that:
     \begin{enumerate}
         \item For each $1 \le j \le n$, $R_j = [x_j, x_j + \delta] \times [y_j, y_j + \delta] \subset U$. 
         \item $\bigcup_{j = 1}^n R_j \supset V$. 

src/dg/complex/space.tex

@@ -24,8 +24,8 @@
 
     and the following are equivalent:
     \begin{enumerate}[label=(C\arabic*)]
-        \item $\cf$ is precompact in $H(U; E)$. 
-        \item $\cf$ is bounded in $H(U; E)$, and for each $x \in U$, $\cf(x) = \bracs{f(x)|f \in \cf}$ is precompact.  
+        \item $\cf$ is relatively compact in $H(U; E)$. 
+        \item $\cf$ is bounded in $H(U; E)$, and for each $x \in U$, $\cf(x) = \bracs{f(x)|f \in \cf}$ is relatively compact.  
     \end{enumerate}
 \end{theorem}
 \begin{proof}

src/dg/derivative/sets.tex

@@ -43,7 +43,7 @@
 
 \begin{definition}
 \label{definition:derivative-garden}
-    Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b \subset 2^E$ be the collection of all finite, precompact, and bounded subsets, respectively, then differentiability with respect to $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b$ correspond to \textbf{Gateaux}, \textbf{Hadamard}, and \textbf{Fréchet} differentiability.
+    Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b \subset 2^E$ be the collection of all finite, relatively compact, and bounded subsets, respectively, then differentiability with respect to $\sigma^E_{\text{Fin}}, \sigma^E_{c}, \sigma^E_b$ correspond to \textbf{Gateaux}, \textbf{Hadamard}, and \textbf{Fréchet} differentiability.
 \end{definition}
 
 
@@ -84,7 +84,7 @@
 \label{proposition:chain-rule-sets-conditions}
     Let $E$, $F$, $G$, be TVSs over $K \in \RC$ with $F, G$ being separated. If $\sigma \subset \mathfrak{B}(E)$ and $\tau \subset \mathfrak{B}(F)$ correspond to the following families of sets on $E$ and $F$:
     \begin{enumerate}
-        \item Precompact sets.
+        \item Relatively compact sets.
         \item Bounded sets.
     \end{enumerate}