Symmetric set

Summary

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 edit

In set notation a subset   of a group   is called symmetric if whenever   then the inverse of   also belongs to   So if   is written multiplicatively then   is symmetric if and only if   where   If   is written additively then   is symmetric if and only if   where  

If   is a subset of a vector space then   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   which happens if and only if   The symmetric hull of a subset   is the smallest symmetric set containing   and it is equal to   The largest symmetric set contained in   is  

Sufficient conditions edit

Arbitrary unions and intersections of symmetric sets are symmetric.

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

Examples edit

In   examples of symmetric sets are intervals of the type   with   and the sets   and  

If   is any subset of a group, then   and   are symmetric sets.

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

See also edit

References edit

  • 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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