# Approximately Additive and Approximately Linear Mappings

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## 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 (*G*_{1}, •), a metric group (*G*_{2},*) with metric *d* and a positive number *η*, suppose that there exists a positive number*ε= ε(η)* such that, if *d*(*f* (*x* • *y*), *f* (*x*) * *f* (*y*)) <*ε* for some *f : G*_{1} *→* *G*_{2} and all *x* and y in G_{1}, then a homomorphism *h: G*_{1} *→ G*_{2} exists with *d*(*f* (*x*), *h*(*x*)) <*η* for all *x* in *G*_{1}. In this case, the equation of homomorphism *h*(*x* • *y*) = *h*(*x*) * *h*(*y*) is called *stable*. Theorem 1.1 with *G*_{1} = *E*_{1}, *G*_{2} = *E*_{2} and with addition as the group operation in each case shows that Cauchy’s equation is stable by this definition with *η= ε*.

## Keywords

Banach Space Functional Equation Additive Mapping Real Banach Space Topological Index## Preview

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