Balanced set

From Encyclopedia of Mathematics
Revision as of 17:26, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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.


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


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


[a1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
How to Cite This Entry:
Balanced set. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article