vanish at infinity

Let $X$ be a locally compact space. A function $f:X\longrightarrow\mathbb{C}$ is said to 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 norm (http://planetmath.org/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 $C_{0}(X)$ .

0.0.1 Remarks

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When $X$ is compact, all functions $X\longrightarrow\mathbb{C}$ vanish at infinity. Hence, $C_{0}(X)=C(X)$ .

Title	vanish at infinity
Canonical name	VanishAtInfinity
Date of creation	2013-03-22 17:50:57
Last modified on	2013-03-22 17:50:57
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	6
Author	asteroid (17536)
Entry type	Definition
Classification	msc 54D45
Classification	msc 54C35
Synonym	zero at infinity
Synonym	vanishes at infinity
Related topic	RegularAtInfinity
Related topic	ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces
Defines	$C_{0}$