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 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 ∪ S ^-1 and S ∩ S ^-1 are symmetric sets.

See also

• Absolutely convex set
• Absorbing set
• Balanced set – Construct in functional analysis
• Bounded set (topological vector space)
• Convex set – In geometry, set that intersects every line into a single line segment
• Minkowski functional
• Star domain

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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• 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.CS1 maint: ref=harv (link)
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)

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