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 – A set that can be "inflated" to eventually always include any given point in a space
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.
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) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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