publication . Preprint . 2005

$q$-norms are really norms

Belbachir, H.; Mirzavaziri, M.; Moslehian, M. S.;
Open Access English
  • Published: 26 Mar 2005
Abstract
Replacing the triangle inequality, in the definition of a norm, by $|x + y| ^{q}\leq 2^{q-1}(|x| ^{q} + |y| ^{q}) $, we introduce the notion of a q-norm. We establish that every q-norm is a norm in the usual sense, and that the converse is true as well.
Subjects
free text keywords: Mathematics - Functional Analysis, 46B20, 46C05
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[1] W. B. Johnson (ed.) and J. Lindenstrauss (ed.), Handbook of the Geometry of Banach Spaces, Vol. 1, North-Holland Publishing Co., Amsterdam, 2001.

[2] S. Saitoh, Generalizations of the triangle inequality, J. Inequal. Pure Appl. Math. 4 (2003), No. 3, Article 62, 5 pp. Hac`ene Belbachir: USTHB/ Facult´e de Math´ematiques, B.P. 32, El Alia, 16111, Bab Ezzouar, Alger, Alg´erie. E-mail address: hbelbachir@usthb.dz Madjid Mirzavaziri: Department of Mathematics, Ferdowsi University, P. O. Box 1159, Mashhad 91775, Iran E-mail address: mirzavaziri@math.um.ac.ir Mohammad Sal Moslehian: Department of Mathematics, Ferdowsi University, P. O. Box 1159, Mashhad 91775, Iran E-mail address: moslehian@ferdowsi.um.ac.ir

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