A Banach–Zarecki Theorem for functions with values in Banach spaces

Abstract

A Banach–Zarecki Theorem for a Banach space-valued function  $F : [0,1] \rightarrow X$ with compact range is presented. We define the strong absolute continuity ($sAC_{||.||_{F}}$) and the bounded variation ($BV_{||.||_{F}}$) of $F$ with respect to the Minkowski functional $||.||_{F}$ associated to the closed absolutely convex hull $C_{F}$ of $F([0,1])$. It is proved that $F$ is $sAC_{||.||_{F}}$ if and only if $F$ is $BV_{||.||_{F}}$, weak continuous on $[0,1]$ and satisfies the weak property $(N)$.