Solution of a problem of Ulam (Q1120092)

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 \(c=c_ 2/(2- 2^{a+b})\).