Norm attaining Lipschitz maps toward vectors

Abstract

We characterize the norm attainment toward vectors for vector-valued Lipschitz maps defined on a general metric space. The main theorem of the present paper states that on a large class of metric spaces including infinite subsets of finite-dimensional spaces, every Lipschitz map attains its norm toward a vector if and only if the range space is finite-dimensional. Furthermore, motivated by the first negative example given by G. Godefroy, some denseness results for norm attaining Lipschitz maps toward vectors are also presented.