On the Hyers-Ulam Stability of Linear Mappings

Let E1 be a real normed vector space and E2 a real Banach space. S. M. Ulam posed the problem: When does a linear mapping near an approximately linear mapping ƒ: E1 ↦ E2 exist? We give a new generalized solution to this problem. An example illustrates when the answer to this question is negative. The behaviour of bounded approximately additive mappings which do not satisfy Hyers-UIam stability is also investigated.