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Bokuan Li
2026-07-07 20:50:19 -04:00
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\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.
Let $X$ be an 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
Let $X$ be an LCH space, then the mapping
\[
E: X \to \Omega(C_0(X)) \quad E(x)(f) = f(x)
\]