On the Maksa-Volkmann functional inequality |f (x + y)| ≥ |f (x) + f (y)| when the range of f is a space of functions

Abstract

P. Volkmann functional inequality |f (x + y)| ≥ |f (x) + f (y)| is extended to functions f : G → F (X, E) where G is an additive group and F (X, E) is the space of functions from a set X to a linear normed space E. As a corollary one proves that an operator T : C (X, K) → C (X, K) which satisfies the functional inequality |T (f + g)| ≥ |T (f) + T (g)| , f, g ∈ C (X, K) is additive. Here we denoted by X a compact topological space, K is R or C and C (X, K) is the linear space of continuous functions defined on X with values in K.