Descriptive characterizations of the integral by seminorms

In this paper, we first define the concept of the limit average range of a function defined on $[0,1]$ and taking values in a Hausdorff locally convex topological vector space (locally convex space) $X$. Then, we present characterizations of the primitive $F:[0,1] \to X$ of an integrable by seminorms function $f:[0,1] \to X$ in terms of the limit average range of $F$. It is shown that the limit average range characterizes the integral by seminorms better then the usual differential.