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, McGraw-Hill Book Company, 1973.

Title	symmetric set
Canonical name	SymmetricSet
Date of creation	2013-03-22 13:48:26
Last modified on	2013-03-22 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 15-00
Classification	msc 46-00