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Solution of a problem of Ulam. (English) Zbl 0672.41027
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

MSC:
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A30 Approximation by other special function classes
Keywords:
Banach space
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References:
[1] Gervirtz, J, Stability of isometries on Banach spaces, (), 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, (), 222-224 · Zbl 0061.26403
[4] Lindenstrauss, J; Szankowski, A, Non-linear perturbations of isometries, (), Palaiseau · Zbl 0585.47007
[5] Ulam, S.M; Ulam, S.M; Ulam, S.M, Sets, numbers, and universes, (1974), MIT Press Cambridge, MA · Zbl 0917.00003
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