Added Fubini's theorem.

src/topology/main/c0.tex

@@ -5,7 +5,7 @@
 \label{definition:vanish-at-infinity}
     Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $f$ \textbf{vanishes at infinity} if for every $U \in \cn_E^o(0)$, $\bracs{f \not\in U}$ is compact.
 
-    The set $C_0(X; E)$ is the space of all functions that vanish at infinity. 
+    The set $C_0(X; E)$ is the space of all functions that vanish at infinity, equipped with the uniform topology. 
 \end{definition}
 
 \begin{proposition}
@@ -38,3 +38,27 @@
 
     so $f \in \ol{C_c(X; E)}$. 
 \end{proof}
+
+\begin{proposition}
+\label{proposition:c0-tensor}
+    Let $X$ be a LCH space and $E$ be a locally convex space over $K \in \RC$. Identify $C_0(X; K) \otimes E$ as a subspace of $C_0(X; E)$ under the natural map
+    \[
+    C_0(X; K) \otimes E \to C_0(X; E) \quad \sum_{j = 1}^n \phi_j \otimes x_j \mapsto \sum_{j = 1}^n x_j \cdot \phi_j
+    \]
+
+    then $C_0(X; K) \otimes E$ is a dense subspace of $C_0(X; E)$. 
+\end{proposition}
+\begin{proof}
+    Let $\phi \in C_0(X; E)$. Using \autoref{proposition:c0-properties}, assume without loss of generality that $\phi \in C_c(X; E)$. 
+
+    Since $\supp{\phi}$ is compact, so is $\phi(X)$ by \autoref{proposition:compact-extensions}. Let $U \in \cn_E^o(0)$ be balanced, then there exists $\seqf{y_j} \subset E \setminus \bracs{0}$ such that $\bigcup_{j = 1}^n (y_j + U) \supset \phi(X)$. For each $1 \le j \le n$, let $V_j = \phi^{-1}(y_j + U)$, then $\seqf{V_j}$ is an open cover of $\supp{\phi}$ consisting of precompact open sets. By \autoref{proposition:lch-partition-of-unity}, there exists a partition of unity $\seqf{\phi_j} \subset C_c(X; [0, 1])$ on $\supp{\phi}$ subordinate to $\seqf{V_j}$. For any $x \in E$, 
+    \begin{align*}
+        \phi(x) - \sum_{j = 1}^n y_j \phi_j(x) &= \sum_{j = 1}^n \phi(x) \phi_j(x) - \sum_{j = 1}^n y_j \phi_j(x) \\
+        &= \sum_{j = 1}^n \phi_j(x)[\phi(x) - y_j] \in \sum_{j = 1}^n \phi_j(x)U \subset U
+    \end{align*}
+    
+    Therefore $(\phi - \sum_{j = 1}^n y_j \phi_j)(X) \subset U$.
+
+\end{proof}
+
+

src/topology/uniform/uc.tex

@@ -163,3 +163,15 @@
     \]
 
 \end{proof}
+
+
+\begin{proposition}
+\label{proposition:uniform-continuous-compact}
+    Let $X$ be a compact uniform space, $Y$ be a uniform space, and $f \in C(X; Y)$, then $f \in UC(X; Y)$.
+\end{proposition}
+\begin{proof}
+    Let $V$ be an entourage of $Y$. For each $x \in X$, let $U_x$ be an entourage of $X$ such that $(f(y), f(z)) \in V$ for all $y, z \in (U_x \circ U_x)(x)$. Since $X$ is compact, there exists $\seqf{x_j} \subset X$ such that $X = \bigcup_{j = 1}^nU_{x_j}(x_j)$. 
+    
+    Let $U = \bigcap_{j = 1}^n U_{x_j}$, then for any $(x, y) \in U$, there exists $1 \le j \le n$ such that $x \in U_{x_j}(x_j)$. In which case, $x, y \in (U_{x_j} \circ U_{x_j})(x_j)$, so $(f(x), f(y)) \in V$. 
+\end{proof}
+