# 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. 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. 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