vanish at infinity
Let $X$ be a locally compact space. A function $f:X\u27f6\u2102$ is said to vanish at infinity if, for every $\u03f5>0$, there is a compact set $K\subseteq X$ such that $$ for every $x\in XK$, where $\parallel \cdot \parallel $ denotes the standard norm (http://planetmath.org/Norm2) on $\u2102$.
If $X$ is noncompact, let $X\cup \{\mathrm{\infty}\}$ be the onepoint compactification of $X$. The above definition can be rephrased as: The extension^{} of $f$ to $X\cup \{\mathrm{\infty}\}$ satisfying $f(\mathrm{\infty})=0$ is continuous at the point $\mathrm{\infty}$.
The set of continuous functions^{} $X\u27f6\u2102$ 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 $X\u27f6\u2102$ vanish at infinity. Hence, ${C}_{0}(X)=C(X)$.
Title  vanish at infinity 

Canonical name  VanishAtInfinity 
Date of creation  20130322 17:50:57 
Last modified on  20130322 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  ${C}_{0}$ 