Any two-dimensional normed space is a generalized Day-James space
type: Item
instance of: scholarly article
title: Any two-dimensional normed space is a generalized Day-James space
zbMATH Open document ID: 1283.46012
DOI: 10.1186/1029-242X-2011-2
author: Javier Alonso
published in: Journal of Inequalities and Applications
publication date: 2013-05-24T00:00:00Z
review text: Let \(\Psi\) be the class of continuous and convex functions \(\psi:[0,1]\to\mathbb{R}\) such that \(\psi(0)=\psi(1)=1\) and \(\max\{t,1-t\}\leq\psi(t)\leq1\) for all \(t\in[0,1]\). It was proved by \textit{F. F. Bonsall} and \textit{J. Ducan} [Numerical ranges II. London: Cambridge University Press (1973; Zbl 0262.47001)] that there is a one-to-one correspondence between the class of absolute and normalized norms on \(\mathbb{R}^2\) and the class \(\Psi\). The author proves that there is a one-to-one correspondence of the class of norms on \(\mathbb{R}^2\) and the class of generalized Day-James norms defined by two functions in \(\Psi\). Note that the latter class was introduced by \textit{W. Nilsrakoo} and the reviewer [J. Inequal. Appl. 2006, Article 26265 (2006; Zbl 1104.46007)].
reviewed by: Satit Saejung
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 6165816
zbMATH Keywords: normed space
zbMATH Keywords: Day-James norm
zbMATH Keywords: Birkhoff orthogonality
Wikidata QID: Q59269945
MaRDI profile type: Publication
cites work: Inner products in normed linear spaces
cites work: Orthogonality in linear metric spaces
full work available at URL: 1029-242x-2011-2
OpenAlex ID: W2145909348
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description: scientific article; zbMATH DE number 6165816