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Difference between revisions of "Function vanishing at infinity"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302701.png" /> be a topological space. A real- or complex-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302702.png" /> is said to vanish at infinity if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302703.png" /> there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302704.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302705.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302706.png" />. For non-compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302707.png" />, such a function can be extended to a [[Continuous function|continuous function]] on the one-point compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302708.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f1302709.png" /> (with value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027010.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027011.png" />).
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Let $X$ be a topological space. A real- or complex-valued function on $X$ is said to ''vanish at infinity'' if for each $\epsilon>0$ there is a compact set $K_\epsilon$ such that $|f(x)|<\epsilon$ for all $x \in X \setminus K_\epsilon$. For non-compact $X$, such a function can be extended to a [[continuous function]] on the [[one-point compactification]] $X^* = X \cup \{\star\}$ of $X$ (with value $0$ at $\star$).
  
The algebra of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027012.png" /> vanishing at infinity is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027013.png" />. In many cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027014.png" /> determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027015.png" />, see e.g. [[Banach–Stone theorem|Banach–Stone theorem]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027016.png" /> is compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027017.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027018.png" /> identifies with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130270/f13027019.png" />.
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The algebra of functions on $X$ vanishing at infinity is denoted by $C_0(X)$. In many cases $C_0(X)$ determines $X$, see e.g. [[Banach–Stone theorem]]. If $X$ is compact, $C_0(X) = C(X)$. The space $C_0(X)$ identifies with $\{f \in X^* : f(\star)=0 \}$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Behrends,  "M-structure and the Banach–Stone theorem" , Springer  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Jarosz,  "Perturbations of Banach spaces" , Springer  (1985)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Behrends,  "M-structure and the Banach–Stone theorem" , Springer  (1979)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Jarosz,  "Perturbations of Banach spaces" , Springer  (1985)</TD></TR>
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</table>
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Latest revision as of 20:25, 10 January 2018

Let $X$ be a topological space. A real- or complex-valued function on $X$ is said to vanish at infinity if for each $\epsilon>0$ there is a compact set $K_\epsilon$ such that $|f(x)|<\epsilon$ for all $x \in X \setminus K_\epsilon$. For non-compact $X$, such a function can be extended to a continuous function on the one-point compactification $X^* = X \cup \{\star\}$ of $X$ (with value $0$ at $\star$).

The algebra of functions on $X$ vanishing at infinity is denoted by $C_0(X)$. In many cases $C_0(X)$ determines $X$, see e.g. Banach–Stone theorem. If $X$ is compact, $C_0(X) = C(X)$. The space $C_0(X)$ identifies with $\{f \in X^* : f(\star)=0 \}$.

References

[a1] E. Behrends, "M-structure and the Banach–Stone theorem" , Springer (1979)
[a2] K. Jarosz, "Perturbations of Banach spaces" , Springer (1985)
How to Cite This Entry:
Function vanishing at infinity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_vanishing_at_infinity&oldid=42708
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article