From 2444406ec140e02f1be63fe5551895cefcc36b61 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Thu, 4 Jun 2026 16:31:26 -0400
Subject: [PATCH] Added facts about C_0.

---
 src/op/example/c0.tex    | 35 +++++++++++++++++++++++++++++++++++
 src/op/example/index.tex |  1 +
 src/topology/main/c0.tex |  3 +++
 3 files changed, 39 insertions(+)
 create mode 100644 src/op/example/c0.tex

diff --git a/src/op/example/c0.tex b/src/op/example/c0.tex
new file mode 100644
index 0000000..e49f43c
--- /dev/null
+++ b/src/op/example/c0.tex
@@ -0,0 +1,35 @@
+\section{$C_0(X)$}
+\label{section:vanishing-infinity-algebra}
+
+\begin{definition}[$C_0(X)$]
+\label{definition:vanishing-infinity-algebra}
+    Let $X$ be a LCH space, then $C_0(X; \complex)$ equipped with pointwise operations and the uniform norm is a $C^*$-algebra. 
+\end{definition}
+
+\begin{theorem}
+\label{theorem:vanishing-infinity-multiplicative-functional}
+    Let $X$ be a LCH space, then the mapping
+    \[
+    E: X \to \Omega(C_0(X)) \quad E(x)(f) = f(x)
+    \]
+
+    is a homeomorphism. Under the identification $X = \Omega(C_0(X))$, the Gelfand transform is the identity. 
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem 7.4]{Zhu}}}. ]
+    Let $X^* = X \sqcup \bracs{\infty}$ be the \hyperref[one-point compactification]{definition:alexandroff-compactification} of $X$. For each $\phi \in \Omega(C_0(X))$, let
+    \[
+    \phi^*: BC(X^*) \to \complex \quad f \mapsto \phi(f - f(\infty)) + f(\infty)
+    \]
+
+    then for each $f, g \in BC(X^*)$, 
+    \begin{align*}
+        fg &= (f - f(\infty))(g - g(\infty)) + f(\infty)(g - g(\infty)) + g(\infty)(f - f(\infty)) + f(\infty)g(\infty) \\
+        \phi^*(fg) &= \phi(f - f(\infty))\phi(g - g(\infty)) + f(\infty)\phi(g - g(\infty)) \\
+        &+ g(\infty)(f - f(\infty)) + f(\infty)g(\infty) \\
+        &= \braksn{\phi(f - f(\infty)) + f(\infty)}\braksn{\phi(g - g(\infty)) + g(\infty)} \\
+        &= \phi^*(f)\phi^*(g)
+    \end{align*}
+
+    so $\phi^* \in \Omega(BC(X^*))$. By \autoref{theorem:multiplicative-functional-bc}, there exists $x \in X^*$ such that $\phi^*(f) = f(x)$ for all $f \in BC(X^*)$. Since $\phi \ne 0$, $x \in X$, and $\phi = E(x)$. 
+\end{proof}
+
diff --git a/src/op/example/index.tex b/src/op/example/index.tex
index 355c0a6..9a12f24 100644
--- a/src/op/example/index.tex
+++ b/src/op/example/index.tex
@@ -3,6 +3,7 @@
 
 \input{./matrix.tex}
 \input{./bounded.tex}
+\input{./c0.tex}
 \input{./hardy.tex}
 \input{./disk.tex}
 \input{./convolution.tex}
diff --git a/src/topology/main/c0.tex b/src/topology/main/c0.tex
index 7230112..5e14ac5 100644
--- a/src/topology/main/c0.tex
+++ b/src/topology/main/c0.tex
@@ -1,6 +1,9 @@
 \section{Continuous Functions Vanishing at Infinity}
 \label{section:vanish-at-infinity}
 
+The following section concerns the properties of spaces of vector valued functions vanishing at infinity. 
+For details regarding the complex-valued cased, in particular its properties as an algebra, see 
+
 \begin{definition}[Vanish at Infinity]
 \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.