Symmetric set

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

S=S^{{-1}}

where $S^{{-1}}=\{x^{{-1}}:x\in S\}$ . In other words, S is symmetric if $x^{{-1}}\in S$ whenever $x\in S$ .

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 $S=-S=\{-x:x\in S\}$ .

Examples

In R, examples of symmetric sets are intervals of the type $(-k,k)$ with $k>0$ , and the sets Z and $\{-1,1\}$ .
Any vector subspace in a vector space is a symmetric set.
If S is any subset of a group, then $SS^{{-1}}$ and $S^{{-1}}S$ are symmetric sets.

References

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

This article incorporates material from symmetric set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

This article is issued from Wikipedia - version of the 1/10/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.