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

Description

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