Symmetric set

In set notation a subset $S$ of a group $G$ is called {{em|symmetric}} if whenever $s\; \backslash in\; S$ 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 $S^\{-1\}\; :=\; \backslash left\backslash \{\; s^\{-1\}\; :\; s\; \backslash in\; S\; \backslash right\backslash \}.$

If $G$ is written additively then $S$ is symmetric if and only if $S\; =\; -\; S$ where $-\; S\; :=\; \backslash \{-\; s\; :\; s\; \backslash in\; S\backslash \}.$

If $S$ is a subset of a vector space then $S$ is said to be a {{em|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 $-\; S\; \backslash subseteq\; S.$

The {{em|symmetric hull}} of a subset $S$ is the smallest symmetric set containing $S,$ and it is equal to $S\; \backslash cup\; -\; S.$ The largest symmetric set contained in $S$ is $S\; \backslash cap\; -\; S.$

Arbitrary unions and intersections of symmetric sets are symmetric.

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

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

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

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

• {{annotated link|Absolutely convex set}}
• {{annotated link|Absorbing set}}
• {{annotated link|Balanced function}}
• {{annotated link|Balanced set}}
• {{annotated link|Bounded set (topological vector space)}}
• {{annotated link|Convex set}}
• {{annotated link|Minkowski functional}}
• {{annotated link|Star domain}}

{{refbegin}}

• R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
• {{Rudin Walter Functional Analysis|edition=2}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}

{{refend}}

{{PlanetMath attribution|id=4528|title=symmetric set}}

{{Functional analysis}}

{{Linear algebra}}

Category:Group theory

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