View source for Symmetric set

{{Short description|Property of group subsets (mathematics)}}
In [[mathematics]], a nonempty subset {{mvar|S}} of a [[Group (mathematics)|group]] {{mvar|G}} is said to be '''symmetric''' if it contains the [[Inverse element|inverses]] of all of its elements.

== Definition ==

In [[Set-builder notation|set notation]] a subset <math>S</math> of a group <math>G</math> is called {{em|symmetric}} if whenever <math>s \in S</math> then the inverse of <math>s</math> also belongs to <math>S.</math> 
So if <math>G</math> is written multiplicatively then <math>S</math> is symmetric if and only if <math>S = S^{-1}</math> where <math>S^{-1} := \left\{ s^{-1} : s \in S \right\}.</math> 
If <math>G</math> is written additively then <math>S</math> is symmetric if and only if <math>S = - S</math> where <math>- S := \{- s : s \in S\}.</math>

If <math>S</math> is a subset of a [[vector space]] then <math>S</math> is said to be a {{em|symmetric set}} if it is symmetric with respect to the [[additive group]] structure of the vector space; that is, if <math>S = - S,</math> which happens if and only if <math>- S \subseteq S.</math> 
The {{em|symmetric hull}} of a subset <math>S</math> is the smallest symmetric set containing <math>S,</math> and it is equal to <math>S \cup - S.</math> The largest symmetric set contained in <math>S</math> is <math>S \cap - S.</math>

== Sufficient conditions ==

Arbitrary [[Union (set theory)|unions]] and [[Intersection (set theory)|intersections]] of symmetric sets are symmetric.

Any [[Linear subspace|vector subspace]] in a vector space is a symmetric set.

== Examples ==

In <math>\R,</math> examples of symmetric sets are intervals of the type <math>(-k, k)</math> with <math>k > 0,</math> and the sets <math>\Z</math> and <math>(-1, 1).</math>

If <math>S</math> is any subset of a group, then <math>S \cup S^{-1}</math> and <math>S \cap S^{-1}</math> are symmetric sets. 

Any [[Balanced set|balanced subset]] of a real or complex [[vector space]] is symmetric.

==See also==

* {{annotated link|Absolutely convex set}}
* {{annotated link|Absorbing set}}
* {{annotated link|Balanced function}}
* {{annotated link|Balanced set}}
* {{annotated link|Bounded set (topological vector space)}}
* {{annotated link|Convex set}}
* {{annotated link|Minkowski functional}}
* {{annotated link|Star domain}}

== References ==
{{refbegin}}
* R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
{{refend}}
{{PlanetMath attribution|id=4528|title=symmetric set}}

{{Functional analysis}}
{{Linear algebra}}

[[Category:Group theory]]


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