Rassias, John M. Solution of a problem of Ulam. (English) Zbl 0672.41027 J. Approximation Theory 57, No. 3, 268-273 (1989). This paper gives the solution of a problem by Ulam concerning conditions for the existence of a linear mapping near an approximately linear mapping by stating the following Theorem: Let X be a normed linear space with norm \(\| \cdot \|_ 1\) and let Y be a Banach space with norm \(\| \cdot \|_ 2\). Assume in addition that f: \(X\mapsto Y\) is a mapping such that f(t\(\cdot x)\) is continuous in t for each fixed x. If there exist \(a,b,0\leq a+b<1\), and \(c_ 2\geq 0\) such that \(\| f(x+y)- [f(x)+f(y)]\|_ 2\leq c_ 2\cdot \| x\|^ a_ 1\cdot \| y\|^ b_ 1\) for all \(x,y\in X\), then there exists a unique linear mapping L:X\(\mapsto Y\) such that \(\| f(x)-L(x)\|_ 2\leq c\cdot \| x\|_ 1^{a+b}\) for all \(x\in X\), where \(c=c_ 2/(2- 2^{a+b})\). Reviewer: E.Quak Cited in 5 ReviewsCited in 152 Documents MSC: 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A30 Approximation by other special function classes Keywords:Banach space PDF BibTeX XML Cite \textit{J. M. Rassias}, J. Approx. Theory 57, No. 3, 268--273 (1989; Zbl 0672.41027) Full Text: DOI References: [1] Gervirtz, J., Stability of isometries on Banach spaces, (Proc. Amer. Math. Soc., 89 (1983)), 633-636 [2] Gruber, P., Stability of isometries, Trans. Amer. Math. Soc., 245, 263-277 (1978) · Zbl 0393.41020 [3] Hyers, D. H., On the stability of the linear functional equation, (Proc. Nat. Acad. Sci. USA, 27 (1941)), 222-224 · Zbl 0061.26403 [4] Lindenstrauss, J.; Szankowski, A., Non-linear Perturbations of Isometries, (Colloquium in honor of Laurent Schwartz, Vol. I (1985)), Palaiseau · Zbl 0585.47007 [5] Ulam, S. M., Sets, Numbers, and Universes (1974), MIT Press: MIT Press Cambridge, MA · Zbl 0917.00003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.