Symmetric set

Short description: Property of group subsets (mathematics)

In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements.

Definition

In set notation a subset $S$ of a group $G$ is called symmetric if whenever $s\in S$ then the inverse of $s$ also belongs to $S.$ So if $G$ is written multiplicatively then $S$ is symmetric if and only if $S=S^{-1}$ where $S^{-1}:=\{s^{-1}:s\in S\}.$ If $G$ is written additively then $S$ is symmetric if and only if $S=-S$ where $-S:=\{-s:s\in S\}.$

If $S$ is a subset of a vector space then $S$ is said to be a symmetric set if it is symmetric with respect to the additive group structure of the vector space; that is, if $S=-S,$ which happens if and only if $-S\subseteq S.$ The symmetric hull of a subset $S$ is the smallest symmetric set containing $S,$ and it is equal to $S\cup -S.$ The largest symmetric set contained in $S$ is $S\cap -S.$

Sufficient conditions

Arbitrary unions and intersections of symmetric sets are symmetric.

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

Examples

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

If $S$ is any subset of a group, then $S\cup S^{-1}$ and $S\cap S^{-1}$ are symmetric sets.

Any balanced subset of a real or complex vector space is symmetric.

See also

• Absolutely convex set
• Absorbing set – Set that can be "inflated" to reach any point
• Balanced set – Construct in functional analysis
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set whose intersection with every line is a single line segment
• Minkowski functional – Function made from a set
• Star domain – Property of point sets in Euclidean spaces

References

• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.