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Symmetric set - Wikipedia

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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.

In set notation a subset $S$ of a group $G$ is called symmetric if whenever $s\in S$ then the inverse of $s$ also belongs to $S.$ So if $G$ is written multiplicatively then $S$ is symmetric if and only if $S=S^{-1}$ where $S^{-1}:=\left\{s^{-1}:s\in S\right\}.$ If $G$ is written additively then $S$ is symmetric if and only if $S=-S$ where $-S:=\{-s:s\in S\}.$

If $S$ is a subset of a vector space then $S$ 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 $S=-S,$ which happens if and only if $-S\subseteq S.$ The symmetric hull of a subset $S$ is the smallest symmetric set containing $S,$ and it is equal to $S\cup -S.$ The largest symmetric set contained in $S$ is $S\cap -S.$

Sufficient conditions

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Arbitrary unions and intersections of symmetric sets are symmetric.

Any vector subspace in a vector space is a symmetric set.

In $\mathbb {R} ,$ examples of symmetric sets are intervals of the type $(-k,k)$ with $k>0,$ and the sets $\mathbb {Z}$ and $(-1,1).$

If $S$ is any subset of a group, then $S\cup S^{-1}$ and $S\cap S^{-1}$ are symmetric sets.

Any balanced subset of a real or complex vector space is symmetric.

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

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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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