Some results on Singer orthogonality and characterizations of inner product spaces
type: Item
instance of: scholarly article
title: Some results on Singer orthogonality and characterizations of inner product spaces
zbMATH Open document ID: 0781.46022
DOI: 10.1007/BF01207467
author: Javier Alonso
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: Let \(E\) be a real normed linear space. Two vectors \(x,y\in E\) are said to be Singer orthogonal [\textit{I. Singer}, Acad. R. P. Romine. Bul. Sţi. Secţ. Mat. Fiz. 9, 29-42 (1957; Zbl 0088.318)], \((x\perp{}{_ s y})\), if either \(\| x\| \| y\|=0\) or \(\| x/\| x\|+ y/\| y\| \|= \| x/\| x\|-y/ \| y\| \|\). Let \(S\) be the unit sphere of \(E\). The following properties are proved: 1) If \(x,y\in S\), \(\alpha\in \mathbb{R}\) and \(x\perp{}{_ s \alpha}x+y\) then \(|\alpha|\leq (1+\sqrt {2})/2\); 2) For every \(x,y\in E\) there exists a unique \(\beta\in\mathbb{R}\) such that \(x+\beta y\perp{}{_ s x}-\beta y\) (moreover, \(1/\sqrt{2} \leq |\beta| \leq \sqrt{2}\) if \(x,y\in S\)). It is well known that if the norm is induced by an inner product then Singer orthogonality coincides with usual orthogonality and \(|\alpha|\) in property 1) is bounded by 1. Conversely, it is proved that this bound is characteristic of inner product spaces. On the other hand, if \(E\) is an inner product space and \(x,y\in S\) then \(x+y\perp{}{_ sx}-y\). This property is also characteristic of such spaces.
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 446284
zbMATH Keywords: Singer orthogonal
MaRDI profile type: Publication
cites work: Orthogonality in linear metric spaces
cites work: Orthogonality in normed linear spaces
cites work: Orthogonality and Linear Functionals in Normed Linear Spaces
cites work: Some geometric characterizations of inear product spaces
cites work: Singer orthogonality and characterizations of inner product spaces
full work available at URL: https://doi.org/10.1007/bf01207467
OpenAlex ID: W2085542454
description: scientific article; zbMATH DE number 446284