Norm attaining Lipschitz maps toward vectors

TL;DR

This paper characterizes when Lipschitz maps attain their norms toward vectors, showing that non-attainment is linked to the finite-dimensionality of the range space, and provides denseness results for such maps.

Contribution

It extends Godefroy's results to a broader setting, establishing a characterization of norm attainment toward vectors based on the dimensionality of the range space.

Findings

Non-norm attaining Lipschitz maps toward vectors exist mainly in finite-dimensional range spaces.

Denseness results are established for norm attaining Lipschitz maps toward vectors.

The main theorem generalizes previous counterexamples to a wider class of metric spaces.

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 non-norm 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.