# Symmetric set

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

where . In other words, *S* is symmetric if whenever .

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 .

## Examples

- In
**R**, examples of symmetric sets are intervals of the type with , and the sets**Z**and . - Any vector subspace in a vector space is a symmetric set.
- If
*S*is any subset of a group, then and 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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