vanish at infinityVanishAtInfinity2008-02-22 22:38:362008-02-22 23:28:14Definitionzero of a function$C_0$% this is the default PlanetMath preamble. as your knowledge
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Let $X$ be a locally compact space. A function $f:X \longrightarrow \mathbb{C}$ is said to \emph{vanish at infinity} if, for every $\epsilon > 0$, there is a compact set $K \subseteq X$ such that $\|f(x)\|<\epsilon$ for every $x \in X-K$, where $\|\cdot\|$ denotes the standard \PMlinkname{norm}{Norm2} on $\mathbb{C}$.
If $X$ is non-compact, let $X \cup \{\infty\}$ be the one-point compactification of $X$. The above definition can be rephrased as: The extension of $f$ to $X \cup \{\infty\}$ satisfying $f(\infty)=0$ is continuous at the point $\infty$.
The set of continuous functions $X \longrightarrow \mathbb{C}$ that vanish at infinity is an algebra over the complex field and is usually denoted by $C_0(X)$.
\subsubsection{Remarks}
\begin{itemize}
\item When $X$ is compact, all functions $X \longrightarrow \mathbb{C}$ vanish at infinity. Hence, $C_0(X) = C(X)$.
\end{itemize}