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In mathematics, a nonempty subset ''S'' of a [[group (mathematics)|group]] ''G'' is said to be '''symmetric''' if
:<math>S=S^{-1}</math>
where <math>S^{-1} = \{ x^{-1} : x \in S \}</math>. In other words, ''S'' is symmetric if <math>x^{-1} \in S</math> whenever <math>x \in S</math>.


If ''S'' is a subset of a [[vector space]], then ''S'' is said to be symmetric if it is symmetric with respect to the additive group structure of the vector space; that is, if <math>S = -S = \{ -x : x \in S \}</math>.


==Examples==
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* In '''R''', examples of symmetric sets are intervals of the type <math>(-k, k)</math> with <math>k > 0</math>, and the sets '''Z''' and <math>\{ -1, 1 \}</math>.
* Any vector subspace in a vector space is a symmetric set.
* If ''S'' is any subset of a group, then <math>SS^{-1}</math> and <math>S^{-1}S</math> are symmetric sets.
 
==References==
*R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
*W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
 
{{PlanetMath attribution|id=4528|title=symmetric set}}
 
{{Functional Analysis}}
 
[[Category:Set theory]]
 
{{settheory-stub}}

Revision as of 12:40, 18 February 2014


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