The mapping \(\gamma_{x,y}\) in normed linear spaces and applications
type: Item
instance of: scholarly article
title: The mapping \(\gamma_{x,y}\) in normed linear spaces and applications
zbMATH Open document ID: 0889.46013
author: Silvestru Sever Dragomir
author: J. J. Koliha
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-10T00:00:00Z
review text: In an arbitrary normed space one can define upper and lower semi-inner products by: \[ (y,x)_i:=\lim_{t\to 0^-}(|x+ty|^2-|x|^2)/2t\text{ and } (y,x)_s:=\lim_{t\to 0^+}(|x+ty|^2-|x|^2)/2t. \] [see, for example, \textit{D. Amir}, ``Characterizations of inner product spaces'' (1986; Zbl 0617.46030)]. The authors begin with a long list of the properties of these functions. For each pair of linearly independent vectors \(x,y\) the authors define a function \(\gamma_{x,y}(t):=(|x+2ty|-|x+ty|)/t\) on \(\mathbb{R}\setminus\{0\}\). Among many properties, the authors show that \(\gamma_{x,y}\) is continuous and increasing; that \(|x|\lim_{t\to 0^-}\gamma_{x,y}(t)=(y,x)_i\); \(|x|\lim_{t\to 0^+}\gamma_{x,y}(t)=(y,x)_s\); and that \(\lim_{t\to \pm\infty}\gamma_{x,y}(t)=\pm|y|\). These relationships are used to show that \(x\) and \(y\) are orthogonal (in the Birkhoff sense) if and only if \(\gamma_{x,y}(u) \leq 0 \leq \gamma_{x,y}(t)\) for all \(u,t\) with \(u < 0 < t\). The function \(\gamma_{x,y}(t)\) is also of interest in inner-product spaces and there it is shown to be twice differentiable.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46C50
Mathematics Subject Classification ID: 46C15
zbMATH DE Number: 1046945
zbMATH Keywords: Birkhoff orthogonality
zbMATH Keywords: inner product space
zbMATH Keywords: upper and lower semi-inner products
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5413
OpenAlex ID: W1965601229
DOI: 10.1006/JMAA.1997.5413
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description: scientific article; zbMATH DE number 1046945