## 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:

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