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Symmetric set
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<h2 id="definition">Definition</h2> <p>In <a href="/facts/Set-builder_notation/0nVHQF6E">set notation</a> a subset S {\displaystyle S} of a group G {\displaystyle G} is called symmetric if whenever s ∈ S {\displaystyle s\in S} then the inverse of s {\displaystyle s} also belongs to S . {\displaystyle S.} So if G {\displaystyle G} is written multiplicatively then S {\displaystyle S} is symmetric if and only if S = S − 1 {\displaystyle S=S^{-1}} where S − 1 := { s − 1 : s ∈ S } . {\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.} If G {\displaystyle G} is written additively then S {\displaystyle S} is symmetric if and only if S = − S {\displaystyle S=-S} where − S := { − s : s ∈ S } . {\displaystyle -S:=\{-s:s\in S\}.} </p><p>If S {\displaystyle S} is a subset of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> then S {\displaystyle S} is said to be a symmetric set if it is symmetric with respect to the <a href="/facts/Additive_group/djWVpp9D">additive group</a> structure of the vector space; that is, if S = − S , {\displaystyle S=-S,} which happens if and only if − S ⊆ S . {\displaystyle -S\subseteq S.} The symmetric hull of a subset S {\displaystyle S} is the smallest symmetric set containing S , {\displaystyle S,} and it is equal to S ∪ − S . {\displaystyle S\cup -S.} The largest symmetric set contained in S {\displaystyle S} is S ∩ − S . {\displaystyle S\cap -S.} </p> <h2 id="sufficient-conditions">Sufficient conditions</h2> <p>Arbitrary <a href="/facts/Union_(set_theory)/KVVXKfWl">unions</a> and <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersections</a> of symmetric sets are symmetric. </p><p>Any <a href="/facts/Linear_subspace/2wtKTgHL">vector subspace</a> in a vector space is a symmetric set. </p> <h2 id="examples">Examples</h2> <p>In R , {\displaystyle \mathbb {R} ,} examples of symmetric sets are intervals of the type ( − k , k ) {\displaystyle (-k,k)} with k > 0 , {\displaystyle k>0,} and the sets Z {\displaystyle \mathbb {Z} } and ( − 1 , 1 ) . {\displaystyle (-1,1).} </p><p>If S {\displaystyle S} is any subset of a group, then S ∪ S − 1 {\displaystyle S\cup S^{-1}} and S ∩ S − 1 {\displaystyle S\cap S^{-1}} are symmetric sets. </p><p>Any <a href="/facts/Balanced_set/Y2ar7E6g">balanced subset</a> of a real or complex <a href="/facts/Vector_space/M1pxMLx2">vector space</a> is symmetric. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Absolutely_convex_set/rJ5HLWkg">Absolutely convex set</a> – Convex and balanced set</li> <li><a href="/facts/Absorbing_set/GCtjsUuX">Absorbing set</a> – Set that can be "inflated" to reach any point</li> <li><a href="/facts/Balanced_function/Y2ar7E6g">Balanced function</a> – Construct in functional analysisPages displaying short descriptions of redirect targets</li> <li><a href="/facts/Balanced_set/Y2ar7E6g">Balanced set</a> – Construct in functional analysis</li> <li><a href="/facts/Bounded_set_(topological_vector_space)/bTsWeU9g">Bounded set (topological vector space)</a> – Generalization of boundedness</li> <li><a href="/facts/Convex_set/vdAuJRJl">Convex set</a> – In geometry, set whose intersection with every line is a single line segment</li> <li><a href="/facts/Minkowski_functional/1qoFAjRs">Minkowski functional</a> – Function made from a set</li> <li><a href="/facts/Star_domain/gMTBXf96">Star domain</a> – Property of point sets in Euclidean spaces</li></ul> <ul><li>R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.</li> <li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1991). <a href="https://archive.org/details/functionalanalys00rudi"><i>Functional Analysis</i></a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="/facts/McGraw-Hill_Science%2fEngineering%2fMath/srSyQSsY">McGraw-Hill Science/Engineering/Math</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-054236-5. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/21163277">21163277</a>.</li> <li>Narici, Lawrence; Beckenstein, Edward (2011). <i>Topological Vector Spaces</i>. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li> <li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). <i>Topological Vector Spaces</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li> <li><a href="/facts/Fran%25C3%25A7ois_Tr%25C3%25A8ves/cpjQbAov">Trèves, François</a> (2006) [1967]. <i>Topological Vector Spaces, Distributions and Kernels</i>. Mineola, N.Y.: Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-45352-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/853623322">853623322</a>.</li></ul> <p><i>This article incorporates material from symmetric set on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i> </p>