balanced set

Definition [1, 2, 3, 4] Let V be a vector spaceMathworldPlanetmath over (or ), and let S be a subset of V. If λSS for all scalars λ such that |λ|1, then S is a balanced set in V. The balanced hull of S, denoted by eq(S), is the smallest balanced set containing S.

In the above, λS={λssS}, and || is the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath (in ), or the modulus of a complex numberMathworldPlanetmathPlanetmath (in ).

0.0.1 Examples and properties

  1. 1.

    Let V be a normed space with norm ||||. Then the unit ball {vV||v||1} is a balanced set.

  2. 2.

    Any vector subspace is a balanced set. Thus, in 3, lines and planes passing through the origin are balanced sets.

0.0.2 Notes

A balanced set is also sometimes called circled [3]. The term balanced evelope is also used for the balanced hull [2]. Bourbaki uses the term équilibré [2], c.f. eq(A) above. In [5], a balanced set is defined as above, but with the condition |λ|=1 instead of |λ|1.


  • 1 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
  • 2 R.E. Edwards, Functional Analysis: Theory and Applications, Dover Publications, 1995.
  • 3 J. Horváth, Topological Vector SpacesMathworldPlanetmath and Distributions, Addison-Wsley Publishing Company, 1966.
  • 4 R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
  • 5 M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition, Academic Press, 1980.
Title balanced set
Canonical name BalancedSet
Date of creation 2013-03-22 15:33:16
Last modified on 2013-03-22 15:33:16
Owner matte (1858)
Last modified by matte (1858)
Numerical id 5
Author matte (1858)
Entry type Definition
Classification msc 46-00
Related topic AbsorbingSet
Defines balanced subset
Defines balanced hull
Defines balanced evelope
Defines circled
Defines équilibré