Generalized Cauchy functional equation and characterizations of inner product spaces
type: Item
instance of: scholarly article
title: Generalized Cauchy functional equation and characterizations of inner product spaces
zbMATH Open document ID: 0755.39007
DOI: 10.1007/BF01835700
author: Damjan Kobal
author: Peter Šemrl
published in: Aequationes Mathematicae
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/137461
review text: A typical result is the following. Let \(X\) be a real or complex linear vector space and \(F\) be \(\mathbb{R}\) or \(\mathbb{C}\). Suppose that \(a_ j\in F\setminus\{0\}\), \(b_ j\in F\), \(c_ j\in F\) (\(j=1,2,3,4\)) satisfy \(\sum_{j=1}^ n a_ j| b_ j|^ 2=\sum_{j=1}^ n a_ j| c_ j|^ 2=0\), \(\sum_{j=1}^ n a_ j b_ j\bar c_ j=\sum_{j=1}^ n a_ j \bar b_ j c_ j=0\), \(b_ j c_ k\neq b_ k c_ j\) (\(j\neq k=1,2,\dots,n\)) and that \(\|\cdot\|: X\to F\) satisfies \(\sum_{j=1}^ n a_ j\| b_ j x+c_ j y\|^ 2=0\) (\(x,y\in X\)). Then \(X\) is an inner product space.
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 92519
zbMATH Keywords: Cauchy functional equation
zbMATH Keywords: linear vector spaces
zbMATH Keywords: inner product spaces
zbMATH Keywords: quadratic functional equations
MaRDI profile type: Publication
cites work: Q5514010
cites work: Orthogonality in normed linear spaces
cites work: Inner Products Characterized by Difference Equations
cites work: Q5345222
cites work: Q5525643
cites work: Q3333552
cites work: Some Functional Equations in Banach Algebras and an Application
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description: scientific article; zbMATH DE number 92519