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

摘要

A Banach-Zarecki Theorem for a Banach space-valued function F: [0, 1] → 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).