Added facts about C_0.

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Bokuan Li
2026-06-04 16:31:26 -04:00
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\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.