Symmetric set

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 {\displaystyle S} of a group G {\displaystyle G} is called symmetric if whenever s ∈ S {\displaystyle s\in S} then the inverse of s {\displaystyle s} also belongs to S . {\displaystyle S.} So if G {\displaystyle G} is written multiplicatively then S {\displaystyle S} is symmetric if and only if S = S − 1 {\displaystyle S=S^{-1}} where S − 1 := { s − 1 : s ∈ S } . {\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.} If G {\displaystyle G} is written additively then S {\displaystyle S} is symmetric if and only if S = − S {\displaystyle S=-S} where − S := { − s : s ∈ S } . {\displaystyle -S:=\{-s:s\in S\}.}

If S {\displaystyle S} is a subset of a vector space then S {\displaystyle 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 , {\displaystyle S=-S,} which happens if and only if − S ⊆ S . {\displaystyle -S\subseteq S.} The symmetric hull of a subset S {\displaystyle S} is the smallest symmetric set containing S , {\displaystyle S,} and it is equal to S ∪ − S . {\displaystyle S\cup -S.} The largest symmetric set contained in S {\displaystyle S} is S ∩ − S . {\displaystyle 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 R , {\displaystyle \mathbb {R} ,} examples of symmetric sets are intervals of the type ( − k , k ) {\displaystyle (-k,k)} with k > 0 , {\displaystyle k>0,} and the sets Z {\displaystyle \mathbb {Z} } and ( − 1 , 1 ) . {\displaystyle (-1,1).}

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

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

See also

  • Absolutely convex set – convex and balanced setPages displaying wikidata descriptions as a fallback
  • Absorbing set – Set that can be "inflated" to reach any point
  • Balanced function – Construct in functional analysisPages displaying short descriptions of redirect targets
  • 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 (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.

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

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