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 $s^{-1}\in S$ whenever s ∈ 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 $S=-S=\{-x:x\in S\}$.

Sufficient conditions

• Arbitrary unions of symmetric sets are symmetric.

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.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

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