Property of group subsets (mathematics)
Inmathematics , a nonempty subsetS of agroup G is said to besymmetric if it contains theinverses of all of its elements.
Inset notation a subsetS {\displaystyle S} G {\displaystyle G} symmetric if whenevers ∈ S {\displaystyle s\in S} s {\displaystyle s} S . {\displaystyle S.} G {\displaystyle G} S {\displaystyle S} S = S − 1 {\displaystyle S=S^{-1}} S − 1 := { s − 1 : s ∈ S } . {\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.} G {\displaystyle G} S {\displaystyle S} S = − S {\displaystyle S=-S} − S := { − s : s ∈ S } . {\displaystyle -S:=\{-s:s\in S\}.}
IfS {\displaystyle S} vector space thenS {\displaystyle S} symmetric set if it is symmetric with respect to theadditive group structure of the vector space; that is, ifS = − S , {\displaystyle S=-S,} − S ⊆ S . {\displaystyle -S\subseteq S.} symmetric hull of a subsetS {\displaystyle S} S , {\displaystyle S,} S ∪ − S . {\displaystyle S\cup -S.} S {\displaystyle S} S ∩ − S . {\displaystyle S\cap -S.}
Sufficient conditions [ edit ] Arbitraryunions andintersections of symmetric sets are symmetric.
Anyvector subspace in a vector space is a symmetric set.
InR , {\displaystyle \mathbb {R} ,} ( − k , k ) {\displaystyle (-k,k)} k > 0 , {\displaystyle k>0,} Z {\displaystyle \mathbb {Z} } ( − 1 , 1 ) . {\displaystyle (-1,1).}
IfS {\displaystyle S} S ∪ S − 1 {\displaystyle S\cup S^{-1}} S ∩ S − 1 {\displaystyle S\cap S^{-1}}
Anybalanced subset of a real or complexvector space is symmetric.
R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977. Rudin, Walter (1991).Functional Analysis 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 .This article incorporates material from symmetric set onPlanetMath , which is licensed under theCreative Commons Attribution/Share-Alike License .
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