On the Stability of the Linear Mapping in Banach Spaces

Abstract

Let <!-- MATH ${E_1},{E_2}$ --> be two Banach spaces, and let <!-- MATH $f:{E_1} \to {E_2}$ --> be a mapping, that is ``approximately linear". S. M. Ulam posed the problem: ``Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem.