Weak derivatives and metric differentiability almost everywhere

Abstract

It is known that a Lipschitz continuous map from the Euclidean domain to a metric space is metrically differentiable almost everywhere. When the metric space is a Banach space dual to separable, the metric differential has its linear counterpart -- weak* differential. However, for an arbitrary metric or Banach space, a Lipschitz map is not necessarily weak* differentiable. This paper introduces an approach based on a concept of weak weak* derivatives. This framework yields a linear representation for the metric differential, allowing for its calculation as the norm of an associated linear operator.