Approximately Additive and Approximately Linear Mappings

Abstract

We have mentioned the concept of stability and given an example for the Cauchy functional equation (1.1). A definition of stability in the case of homomorphisms between metric groups was suggested by a problem posed by S.M. Ulam in 1940 (see Ulam (1960), p. 64). Given a group (G₁, •), a metric group (G₂,*) with metric d and a positive number η, suppose that there exists a positive numberε= ε(η) such that, if d(f (x • y), f (x) * f (y)) <ε for some f : G₁ → G₂ and all x and y in G₁, then a homomorphism h: G₁ → G₂ exists with d(f (x), h(x)) <η for all x in G₁. In this case, the equation of homomorphism h(x • y) = h(x) * h(y) is called stable. Theorem 1.1 with G₁ = E₁, G₂ = E₂ and with addition as the group operation in each case shows that Cauchy’s equation is stable by this definition with η= ε.