On functionals which are orthogonally additive modulo \(\mathbb{Z}\)
type: Item
instance of: scholarly article
title: On functionals which are orthogonally additive modulo \(\mathbb{Z}\)
zbMATH Open document ID: 0862.39012
DOI: 10.1007/BF03322177
published in: Results in Mathematics
publication date: 1997-05-25T00:00:00Z
review text: The Cauchy difference \(C_f(x,y):=f(x+y)-f(x)-f(y)\) is studied for functions \(f\) mapping a real inner product space \(E\) \((\dim E>1)\) into the reals. The functions satisfying the congruence \(C_f(x,y)\in\mathbb{Z}\) (the set of integers) for all orthogonal \(x,y\in E\) are characterized by means of the square of the norm in \(E\) and the unique real linear functional acting on \(E\). This is an analogue, obtained both for sets with Baire property and for Christensen measurable sets, of a result by \textit{K. Baron} and \textit{G. L. Forti} [Result. Math. 26, No. 3-4, 205-210 (1994; Zbl 0828.39010)]. Following the pattern from the paper just quoted, the author also supplies solutions \(F:E\to \mathbb{C}\) of the conditional Cauchy equation \(F(x+y)=F(x)F(y)\) for all orthogonal \(x,y\in E\).
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 933253
zbMATH Keywords: orthogonally additive linear functionals
zbMATH Keywords: Cauchy difference
zbMATH Keywords: inner product space
zbMATH Keywords: congruence
zbMATH Keywords: Christensen measurable sets
zbMATH Keywords: conditional Cauchy equation
author: Janusz Brzdȩk
reviewed by: Bogdan A. Choczewski
MaRDI profile type: Publication
cites work: Orthogonality and additivity modulo \(\mathbb{Z}\)
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cites work: On orthogonally additive mappings
full work available at URL: https://doi.org/10.1007/bf03322177
OpenAlex ID: W2010473175
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description: scientific article; zbMATH DE number 933253