vanish at infinity

Let X be a locally compact space. A function f:X is said to vanish at infinity if, for every ϵ>0, there is a compact set KX such that |f(x)|<ϵ for every xX-K, where denotes the standard norm ( on .

If X is non-compact, let X{} be the one-point compactification of X. The above definition can be rephrased as: The extensionPlanetmathPlanetmath of f to X{} satisfying f()=0 is continuous at the point .

The set of continuous functionsMathworldPlanetmath X that vanish at infinity is an algebra over the complex field and is usually denoted by C0(X).

0.0.1 Remarks

  • When X is compact, all functions X vanish at infinity. Hence, C0(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 C0