Descriptive characterizations of the integral by seminorms

TL;DR

This paper introduces the concept of the limit average range for functions in locally convex spaces and demonstrates its effectiveness in characterizing integrals by seminorms, surpassing traditional differential methods.

Contribution

It defines the limit average range and shows it provides a more accurate characterization of integrals by seminorms in locally convex spaces.

Findings

Limit average range characterizes the integral by seminorms.

The approach improves upon traditional differential characterizations.

Provides new insights into integration in locally convex spaces.

Abstract

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.