Symmetric set

In mathematics, a nonempty subset S of a group G is said to be symmetric if

where . In other words, S is symmetric if whenever .

If S is a subset of a vector space, then S is said to be symmetric if it is symmetric with respect to the additive group structure of the vector space; that is, if .


  • In R, examples of symmetric sets are intervals of the type with , and the sets Z and .
  • Any vector subspace in a vector space is a symmetric set.
  • If S is any subset of a group, then and are symmetric sets.


  • R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
  • W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.

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