## 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

Banach space
Full Text:

### 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.