Abstract
In this note we prove the following: Let n ≥ 2 be a fixed integer. A system of additive functions \({A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}\) is linearly dependent (as elements of the \({\mathbb{R}}\) vector space \({\mathbb{R}^{\mathbb{R}}}\)), if and only if, there exists an indefinite quadratic form \({Q:\mathbb{R}^{n}\to\mathbb{R} }\) such that \({Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}\) or \({Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}\) holds for all \({x\in\mathbb{R}}\) .
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Aczél J.: Lectures on functional equations and their applications. Academic Press, New York (1966)
Maksa, Gy., Rätz, J.: /Remark 5./, Proceedings of the Nineteenth International Symposium on Functional Equations, Centre for Information Theory, University of Waterloo, Waterloo, Ontario, Canada, N2L3G1, p. 56
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This work was supported by the Hungarian Scientific Research Fund (OTKA), Grant No. NK 81402.
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Kocsis, I. On the Linear Dependence of a Finite Set of Additive Functions. Results. Math. 62, 67–71 (2012). https://doi.org/10.1007/s00025-011-0130-0
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DOI: https://doi.org/10.1007/s00025-011-0130-0