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]- Absolutely convex set – convex and balanced set
- Absorbing set – Set that can be "inflated" to reach any point
- Balanced function – Construct in functional analysis
- 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
[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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