balanced set

Definition [1,2,3,4] Let $V$ be a vector space over $\sR$ (or $\sC$ ), 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 $\operatorname{eq}(S)$ , 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 $\sR$ ), or the modulus of a complex number (in $\sC$ ).

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 $\sR^3$ , lines and planes passing through the origin are balanced sets.

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 $|\lambda|\le 1$ .

Bibliography

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.