symmetric set
Definition A subset $A$ of a group $G$ is said to be symmetric^{} if $A={A}^{1}$, where ${A}^{1}=\{{a}^{1}:a\in A\}$. In other , $A$ is symmetric if ${a}^{1}\in A$ whenever $a\in A$.
If $A$ is a subset of a vector space^{}, then $A$ is said to be symmetric if it is symmetric with respect to the additive group^{} structure^{} of the vector space; that is, if $A=\{a:a\in A\}$ [1].
0.0.1 Examples

1.
In $\mathbb{R}$, examples of symmetric sets are intervals of the type $(k,k)$ with $k>0$, and the sets $\mathbb{Z}$ and $\{1,1\}$.

2.
Any vector subspace in a vector space is a symmetric set.

3.
If $A$ is any subset of a group, then $A\cap {A}^{1}$ and $A\cup {A}^{1}$ are symmetric sets.
References
 1 R. Cristescu, Topological vector spaces^{}, Noordhoff International Publishing, 1977.
 2 W. Rudin, Functional Analysis^{}, McGrawHill Book Company, 1973.
Title  symmetric set 

Canonical name  SymmetricSet 
Date of creation  20130322 13:48:26 
Last modified on  20130322 13:48:26 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  7 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 20A99 
Classification  msc 22A05 
Classification  msc 1500 
Classification  msc 4600 