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On the stability of the linear mapping in Banach spaces. (English) Zbl 0398.47040

S. M. Ulam posed the problem: Let E 1 ,E 2 be two Banach spaces, and let f:E 1 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 E 2 be a mapping such that f(tx) is continuous in t for each fixed x. Assume that there exists Θ0 and p[0,1) such that

f(x+y)-f(x)-f(y) x p +y p Θ,

for any x,y. The there exists a unique linear mapping T:E 1 E 2 such that f(x)-T(x) x p 2Θ 2-2 p , for any xE 1 .

Reviewer: Th. M. Rassias

47H14Perturbations of nonlinear operators
47A55Perturbation theory of linear operators
46B99Normed linear spaces and Banach spaces