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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 $\Vert \cdot \Vert\sb 1$ and let Y be a Banach space with norm $\Vert \cdot \Vert\sb 2$. 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 $a,b,0\le a+b<1$, and $c\sb 2\ge 0$ such that $\Vert f(x+y)- [f(x)+f(y)]\Vert\sb 2\le c\sb 2\cdot \Vert x\Vert\sp a\sb 1\cdot \Vert y\Vert\sp b\sb 1$ for all $x,y\in X$, then there exists a unique linear mapping L:X$\mapsto Y$ such that $\Vert f(x)-L(x)\Vert\sb 2\le c\cdot \Vert x\Vert\sb 1\sp{a+b}$ for all $x\in X$, where $c=c\sb 2/(2- 2\sp{a+b})$.
Reviewer: E.Quak

41A65Abstract approximation theory
41A30Approximation by other special function classes
Banach space
Full Text: DOI
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[3] Hyers, D. H.: On the stability of the linear functional equation. Proc. nat. Acad. sci. USA 27, 222-224 (1941) · Zbl 0061.26403
[4] Lindenstrauss, J.; Szankowski, A.: Non-linear perturbations of isometries. Colloquium in honor of Laurent Schwartz (1985) · Zbl 0585.47007
[5] Ulam, S. M.: Sets, numbers, and universes. (1974) · Zbl 0558.00017