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Difference between revisions of "Support of a function"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912801.png" /> defined on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912802.png" />''
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{{TEX|done}}
  
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The smallest closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912803.png" /> such that the values of the numerical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912804.png" /> are zero everywhere on the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912805.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912806.png" /> is the closure of the set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912807.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912808.png" />.
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{{MSC|54A}}
  
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[[Category:General topology]]
  
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Let $X$ be a topological space and $f:X\to \mathbb R$ a function. The support of $f$, denoted by ${\rm supp}\, (f)$ is the smallest closed set outside of which the function $f$ vanishes identically. ${\rm supp}\, (f)$ can also be characterized as
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* the complent of the union of all sets on which $f$ vanishes identically
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* the closure of the set $\{f\neq 0\}$.
  
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====Comments====
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The same concept can be readily extended to maps taking values in a vector space or more generally in an additive group.
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A function is said to be of compact support if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s0912809.png" /> is compact. The functions of compact support with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s09128010.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091280/s09128011.png" /> (or other rings or fields), form a vector space.
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A function $f$ is said to have compact support if ${\rm supp}\, (f)$ is compact. If the target $V$ is a vector space, the set of functions $f:X\to V$ with compact support is also a vector space.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1966)  pp. 38</TD></TR></table>
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{|
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|valign="top"|{{Ref|Ru}}|| W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1966)  pp. 38\, .
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Latest revision as of 14:28, 29 November 2012


2020 Mathematics Subject Classification: Primary: 54A [MSN][ZBL]

Let $X$ be a topological space and $f:X\to \mathbb R$ a function. The support of $f$, denoted by ${\rm supp}\, (f)$ is the smallest closed set outside of which the function $f$ vanishes identically. ${\rm supp}\, (f)$ can also be characterized as

  • the complent of the union of all sets on which $f$ vanishes identically
  • the closure of the set $\{f\neq 0\}$.

The same concept can be readily extended to maps taking values in a vector space or more generally in an additive group.

A function $f$ is said to have compact support if ${\rm supp}\, (f)$ is compact. If the target $V$ is a vector space, the set of functions $f:X\to V$ with compact support is also a vector space.

References

[Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 38\, .
How to Cite This Entry:
Support of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_function&oldid=28948
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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