balanced set
Definition [1, 2, 3, 4] Let $V$ be a vector space^{} over $\mathbb{R}$ (or $\u2102$), and let $S$ be a subset of $V$. If $\lambda S\subset S$ for all scalars $\lambda $ such that $\lambda \le 1$, then $S$ is a balanced set in $V$. The balanced hull of $S$, denoted by $\mathrm{eq}(S)$, is the smallest balanced set containing $S$.
In the above, $\lambda S=\{\lambda s\mid s\in S\}$, and $\cdot $ is the absolute value^{} (in $\mathbb{R}$), or the modulus of a complex number^{} (in $\u2102$).
0.0.1 Examples and properties

1.
Let $V$ be a normed space with norm $\cdot $. Then the unit ball $\{v\in V\mid v\le 1\}$ is a balanced set.

2.
Any vector subspace is a balanced set. Thus, in ${\mathbb{R}}^{3}$, lines and planes passing through the origin are balanced sets.
0.0.2 Notes
References
 1 W. Rudin, Functional Analysis, McGrawHill Book Company, 1973.
 2 R.E. Edwards, Functional Analysis: Theory and Applications, Dover Publications, 1995.
 3 J. Horváth, Topological Vector Spaces^{} and Distributions, AddisonWsley 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  20130322 15:33:16 
Last modified on  20130322 15:33:16 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  5 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 4600 
Related topic  AbsorbingSet 
Defines  balanced subset 
Defines  balanced hull 
Defines  balanced evelope 
Defines  circled 
Defines  équilibré 