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 \((N)\).