Symmetric set

In mathematics, a nonempty subset S of a group G is said to be symmetric if

S=S^{{-1}}

where $S^{{-1}}=\{x^{{-1}}:x\in S\}$ . In other words, S is symmetric if $x^{{-1}}\in S$ whenever $x\in S$ .

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 $S=-S=\{-x:x\in S\}$ .

Examples

In R, examples of symmetric sets are intervals of the type $(-k,k)$ with $k>0$ , and the sets Z and $\{-1,1\}$ .
Any vector subspace in a vector space is a symmetric set.
If S is any subset of a group, then $S\cup S^{-1}$ and $S^{-1}\cap S$ are symmetric sets.

References

R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

This article incorporates material from symmetric set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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