# Balanced set

In linear algebra and related areas of mathematics a **balanced set**, **circled set** or **disk** in a vector space (over a field *K* with an absolute value function | |) is a
set *S* such that for all scalars α with |α| ≤ 1

where

The **balanced hull** or **balanced envelope** for a set *S* is the smallest balanced set containing *S*. It can be constructed as the intersection of all balanced sets containing *S*.

## Examples

- The open and closed balls centered at 0 in a normed vector space are balanced sets.
- Any subspace of a real or complex vector space is a balanced set.
- The cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field
*K*). - Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do. As a result, ℂ and ℝ
^{2}are entirely different as far as their vector space structure is concerned. - If p is a semi-norm on a linear space X, then for any constant c>0, the set {x ∈ X | p(x)≤c} is balanced.

## Properties

- The union and intersection of balanced sets is a balanced set.
- The closure of a balanced set is balanced.
- By definition (not property), a set is absolutely convex if and only if it is convex and balanced

## See also

## References

- Robertson, A.P.; W.J. Robertson (1964).
*Topological vector spaces*. Cambridge Tracts in Mathematics.**53**. Cambridge University Press. p. 4. - W. Rudin (1990).
*Functional Analysis*(2nd ed.). McGraw-Hill, Inc. ISBN 0-07-054236-8. - H.H. Schaefer (1970).
*Topological Vector Spaces*. GTM.**3**. Springer-Verlag. p. 11. ISBN 0-387-05380-8.

This article is issued from Wikipedia - version of the 3/17/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.