# Balanced set

From Encyclopedia of Mathematics

A set in a real or complex vector space such that and imply . An example of a balanced set is given by the unit ball in a normed vector space and, generally, by a neighbourhood of zero in a base of neighbourhoods of zero in a topological vector space. These neighbourhoods of zero are moreover absorbing, i.e. such that for any there exists an such that for . If is a convex, absorbing and balanced set, then the functional is a semi-norm, i.e. it has the properties

A balanced set is also called centred.

#### References

[1] | L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian) |

#### Comments

The functional mentioned above is also called the Minkowski functional of the convex, absorbing and balanced set .

#### References

[a1] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 |

**How to Cite This Entry:**

Balanced set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Balanced_set&oldid=18616

This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article