balanced set

Definition [1, 2, 3, 4] Let $V$ be a vector space over $\mathbb{R}$ (or $\mathbb{C}$ ), and let $S$ be a subset of $V$ . If $\lambda S\subset S$ for all scalars $\lambda$ such that $|\lambda|\leq 1$ , then $S$ is a balanced set in $V$ . The balanced hull of $S$ , denoted by $\operatorname{eq}(S)$ , 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 $\mathbb{C}$ ).

0.0.1 Examples and properties

1.

Let $V$ be a normed space with norm $||\cdot||$ . Then the unit ball $\{v\in V\mid||v||\leq 1\}$ 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

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. $\operatorname{eq}(A)$ above. In [5], a balanced set is defined as above, but with the condition $|\lambda|=1$ instead of $|\lambda|\leq 1$ .

References

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 Spaces 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