an:03618859
Zbl 0398.47040
Rassias, Themistocles M.
On the stability of the linear mapping in Banach spaces
EN
Proc. Am. Math. Soc. 72, 297-300 (1978).
00141188
1978
j
47H14 47A55 46B99
Approximately Linear Mapping
S. M. Ulam posed the problem: Let \(E_1, E_2\) be two Banach spaces, and let \(f: E_1 \to E_2\) be a mapping, that is ``approximately linear''. Give conditions in order for a linear mapping near an approximately linear mapping to exist. The author has given an answer to Ulam's problem. In fact the following theorem has been stated and proved.
Theorem: Consider \(E_1, E_2\) to be two Banach spaces, and let \(f: E_1 \to E_2\) be a mapping such that \(f(tx)\) is continuous in \(t\) for each fixed \(x\). Assume that there exists \(\Theta\geq 0\) and \(p\in[0,1)\) such that
\[
\frac{\| f(x+y)-f(x)-f(y)\|}{\| x\|^p+\| y\|^p}\leq \Theta,
\]
for any \(x,y\in\mathbb R\). The there exists a unique linear mapping \(T: E_1 \to E_2\) such that \(\frac{\| f(x)-T(x)\|}{\| x\|^p}\leq \frac{2\Theta}{2-2^p}\), for any \(x\in E_1\).
Th. M. Rassias