Solution of a problem of Ulam

In this paper we solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist” and establish results involving a product of powers of norms 5.]; 5.]; 5.]]. There has been much activity on a similar “ -isometry” problem of Ulam 1.], 633–636; 2.], 263–277; 4.]]. This work represents an improvement and generalization of the work of 3.], 222–224].