Symmetric set

Template:Short description In mathematics, a nonempty subset Template:Mvar of a group Template:Mvar is said to be symmetric if it contains the inverses of all of its elements.

Definition

In set notation a subset $S$ of a group $G$ is called Template:Em 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 $S^{-1}:=\{s^{-1}:s\in S\}.$ 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 Template:Em 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 Template:Em 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

Arbitrary unions and intersections of symmetric sets are symmetric.

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

Examples

In $ℝ,$ 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.

See also

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References

• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
• Template:Rudin Walter Functional Analysis
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels

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