Symmetric set

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

S = S−1

where S−1 = { s−1 : s S}. In other words, S is symmetric if s−1 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 = { -s : s S}.

Sufficient conditions

  • Arbitrary unions and intersections of symmetric sets are symmetric.

Examples

  • In , examples of symmetric sets are intervals of the type (-k, k) with k > 0, and the sets and { -1, 1 }.
  • Any vector subspace in a vector space is a symmetric set.
  • If S is any subset of a group, then S S−1 and S S−1 are symmetric sets.

See also

References

  • R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
  • Rudin, Walter (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.
  • 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 (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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