Quasi-linear mappings

Abstract

Let E[1], E[2] be real normed spaces and let ε ϵ [0,1). The paper deals with the system of inequalities || f(x + y) - f(x) - f(y)|| ≤ ε min{||f(x + y)||,||f(x) + f(y)||} for x , y ϵ E[1], II f(αx) - αf(x)|| ≤ ε min {||f(αx)||, ||αf(x)||} for x ϵ E[1], α ϵ R, where f maps E[1] into E[2]. We prove that some basic theorems concerning linear operators also hold for mappings satisfying these inequalities. In the next part of the paper we assume additionally that E 2 = R and f is continuous. Then we prove that there exists a continuous linear mapping L : E[1] —> R such that I f(x) - L(x) |≤ ε min {| f(x) |, | L(x) |} for x ϵ E[1] . In the set of such linear mappings there exists a unique one, which is the best linear approximation of f.