On mappings, orthogonally additive in the Birkhoff-James sense
type: Item
instance of: scholarly article
title: On mappings, orthogonally additive in the Birkhoff-James sense
zbMATH Open document ID: 0594.39007
DOI: 10.1007/BF02189914
author: György Szabó
published in: Aequationes Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137115
review text: Let \((X,\| \|)\) be a real normed linear space but not an inner product space of dimension \(\geq 2\) and \((G,+)\) be an arbitrary groupoid. Define the following binary relation \(\perp =\{(x,y)\in X\times X,\| x+\lambda y\| \geq \| x\|\) for all \(\lambda\in {\mathbb{R}}\}\). It is proved that a mapping F:X\(\to G\) is even and satisfies \[ (1)\quad F(x+y)=F(x)+F(y)\quad for\quad al\quad x,y\in X\quad with\quad x\perp y \] if and only if F is of the form \(F(x)=a_ 0\), if \(x=0\) and \(F(x)=a\), if \(x\neq 0\) where \(a_ 0,a\in G\) such that \(a_ 0+a_ 0=a_ 0\), \(a+a=a\) and \(a_ 0+a=a+a_ 0=a\). On the base of this result the author proves that if \((G,+)\) is an abelian group, then every function F:X\(\to G\) which satisfies (1) is additive.
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46C99
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3955623
zbMATH Keywords: additive mappings
zbMATH Keywords: Birkhoff-James orthogonality
zbMATH Keywords: real normed linear space
zbMATH Keywords: groupoid
reviewed by: Marek Cezary Zdun
MaRDI profile type: Publication
Wikidata QID: Q112879241
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cites work: Q3312643
OpenAlex ID: W1998198062
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description: scientific article; zbMATH DE number 3955623