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.