On orthogonality spaces admitting nontrivial even orthogonally additive mappings
type: Item
instance of: scholarly article
title: On orthogonality spaces admitting nontrivial even orthogonally additive mappings
zbMATH Open document ID: 0722.39007
DOI: 10.1007/BF01903720
author: György Szabó
published in: Acta Mathematica Hungarica
publication date: 1990-01-01T00:00:00Z
review text: Let (X,\(\perp)\) be an orthogonality space (X a real vector space and \(\perp\) an orthogonality relation on X). Let \((Y,+)\) be an abelian group. Let \((e)Hom_{\perp}(X,Y)\) denote the set of mappings \(\{\) E: \(X\to Y|\) E is even orthogonally additive\(\}\), i.e., \(E(x+y)=E(x)+E(y)\) for all x,y\(\in X\) with \(x\perp y\), E even. The author proves: If dim \(X\geq 3\) and \((e)Hom_{\perp}(X,Y)\neq \{0\}\), then there exists a functional \(\rho\) : \(X\to {\mathbb{R}}\) with the following properties: (i) \(\rho (0)=0\), \(\rho (x)>0\) for all \(x\in X\setminus \{0\}\); (ii) \(\rho (\lambda x)=| \lambda | \rho (x)\), for all \(x\in X\) and \(\lambda\in {\mathbb{R}}\); (iii) \(E(x)=E(y)\) for any \(E\in (e)Hom_{\perp}(X,Y)\) and every x,y\(\in X\) such that \(\rho (x)=\rho (y).\) The main result: If dim \(X\geq 3\) and \((e)Hom_{\perp}(X,Y)\neq \{0\}\), then there exists an equivalent inner product \(<\cdot,\cdot >: X\times Y\to {\mathbb{R}}\), i.e. \(x\perp y\Leftrightarrow <x,y>=0\) for all x,y\(\in X\), and \((e)Hom_{\perp}(X,Y)=\{a\circ \| \cdot \|^ 2|\) \(a\in Hom({\mathbb{R}},Y)\}\).
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 4190326
zbMATH Keywords: Birkhoff-James orthogonality
zbMATH Keywords: quasi norm
zbMATH Keywords: Jordan-von Neumann functional equations
zbMATH Keywords: Cauchy functional equation
zbMATH Keywords: nontrivial even orthogonally additive mappings
zbMATH Keywords: orthogonality space
zbMATH Keywords: vector space
zbMATH Keywords: abelian group
zbMATH Keywords: inner product
MaRDI profile type: Publication
cites work: Q5530030
cites work: Orthogonally additive and orthogonally increasing functions on vector spaces
cites work: Orthogonality and Linear Functionals in Normed Linear Spaces
cites work: Inner products in normed linear spaces
cites work: On orthogonally additive mappings
cites work: Orthogonality and Nonlinear Functionals on Banach Spaces
cites work: On mappings, orthogonally additive in the Birkhoff-James sense
full work available at URL: https://doi.org/10.1007/bf01903720
OpenAlex ID: W2083863401
description: scientific article; zbMATH DE number 4190326