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Zbl 0672.41027
Rassias, John M.
Solution of a problem of Ulam.
(English)
[J] J. Approximation Theory 57, No.3, 268-273 (1989). ISSN 0021-9045

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 $c=c\sb 2/(2- 2\sp{a+b})$.
[E.Quak]
MSC 2000:
*41A65 Abstract approximation theory
41A30 Approximation by other special function classes

Keywords: Banach space

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