Norm attaining Lipschitz maps toward vectors

Abstract

We extend the recent result of G. Godefroy which concerns the existence of non-norm attaining Lipschitz maps in order to characterize the norm attainment toward vectors for Lipschitz maps in the general setting of underlying space. The main theorem of the present paper states that the existence of nonnorm attaining Lipschitz maps toward vectors on a large class of metric spaces is characterized by the finite-dimensionality of range space. As an extension of his counterexample, some denseness results on norm attaining Lipschitz maps toward vectors are also shown.Definition 1.1. A Lipschitz map f ∈ Lip 0 (M, Y ) is said to attain its norm toward y ∈ Y if there exists a sequence of distinct pairs {(p n , q n )} ⊆ M × M such that