On orthogonally additive mappings. IV
type: Item
instance of: scholarly article
title: On orthogonally additive mappings. IV
zbMATH Open document ID: 0679.39005
DOI: 10.1007/BF01839496
author: György Szabó
author: Jürg Rätz
published in: Aequationes Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137317
review text: [For part II see the first author, Publ. Math. 35, No.3/4, 241-249 (1988; reviewed above).] Let \(\Phi\) denote a field of characteristic \(\neq 2\), X a \(\Phi\)-vector space of dimension \(\geq 2\) and \((Y,+)\) an abelian group. Furthermore let \(\perp\), which is called orthogonality on X, be a binary relation satisfying certain appropriate conditions on X. The mappings A, Q and F: \(X\to Y\) are said to be additive, quadratic, or orthogonally additive (\(\perp\)-additive), if they satisfy the equations: \(A(x+y)=A(x)+A(y),\) \(x,y\in X\), \(Q(x+y)+Q(x-y)=2Q(x)+2Q(y),\) \(x,y\in X\), or \(F(x+y)=F(x)+F(y),\) \(x,y\in X\), \(x\perp y\), respectively. The main result in this paper is to prove three inclusion relations between the five sets \(\{\) \(A|\) A is additive\(\}\), \(\{\) \(Q|\) Q is quadratic\(\}\), \(\{\) \(F|\) F is \(\perp\)-additive\(\}\), \(\{\) \(D|\) D is odd and \(\perp\)-additive\(\}\) and \(\{\) \(E|\) E is even and \(\perp\)- additive\(\}\). The authors close with some results in the case when the abelian group \((Y,+)\) is 2-torsion-free.
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46C99
zbMATH DE Number: 4113190
zbMATH Keywords: orthogonality
zbMATH Keywords: additive
zbMATH Keywords: quadratic
zbMATH Keywords: orthogonally additive
zbMATH Keywords: 2-torsion- free
zbMATH Keywords: vector space
zbMATH Keywords: abelian group
reviewed by: Hiroshi Haruki
MaRDI profile type: Publication
cites work: Q5530030
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cites work: Orthogonally additive and orthogonally increasing functions on vector spaces
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cites work: On orthogonally additive mappings
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cites work: Orthogonality and Nonlinear Functionals on Banach Spaces
cites work: On mappings, orthogonally additive in the Birkhoff-James sense
cites work: On orthogonality spaces admitting nontrivial even orthogonally additive mappings
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description: scientific article; zbMATH DE number 4113190