S. M. Ulam posed the problem: Let be two Banach spaces, and let 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 to be two Banach spaces, and let be a mapping such that is continuous in for each fixed . Assume that there exists and such that
for any . The there exists a unique linear mapping such that , for any .