Generalized Cauchy functional equation and characterizations of inner product spaces. (English) Zbl 0755.39007

Kobal, Damjan; Šemrl, Peter

Aequationes Math. 43, No. 2-3, 183-190 (1992).

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.

Reviewer: J.Aczél (Waterloo / Ontario)

MSC:

39B52	Functional equations for functions with more general domains and/or ranges
46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords:

Cauchy functional equation; linear vector spaces; inner product spaces; quadratic functional equations

References:

[1]	Aczèl, J. Lectures on functional equations and their applications. Academic Press, New York and London, 1966. · Zbl 0139.09301
[2]	Carlsson, S. O. Orthogonality in normed linear spaces. Ark. Mat.4 (1961), 297–318. · Zbl 0107.08803
[3]	Johnson, G. G. Inner products characterized by difference equations. Proc. Amer. Math. Soc.37 (1973), 535–536. · Zbl 0268.46019
[4]	Jordan, P. andvon Neumann, J.,Inner products in linear metric spaces. Annals of Math.36 (1935), 291–302. · JFM 61.0435.05
[5]	Kurepa, S. The Cauchy functional equation and scalar product in vector spaces. Glasnik Mat. Fiz.-Astr.19 (1964), 23–36. · Zbl 0134.32601
[6]	Kurepa, S. Quadratic and sesquilinear functionals. Glasnik Mat. Fiz.-Astr.20 (1965), 79–92. · Zbl 0147.35202
[7]	Rassias, Th. M. New characterizations of inner product spaces. Bull. Sci. Math.108 (1984), 95–99. · Zbl 0544.46016
[8]	Šemrl, P. Additive functions and two characterizations of inner product spaces. To appear in Glasnik Mat. Ser. III.
[9]	Vukman, J.,Some functional equations in Banach algebras and an application. Proc. Amer. Math. Soc.100 (1987), 133–136. · Zbl 0623.46021