On the spaceability of the sets of norm-attaining Lipschitz functions

Abstract

Motivated by the result of Dantas et al. in Nonlinear Anal. (2023) that there exist metric spaces for which the set of strongly norm-attaining Lipschitz functions does not contain an isometric copy of 0, we introduce and study a weaker notion of norm-attainment for Lipschitz functions called the pointwise normattainment. As a main result, we show that for every infinite metric space , there exists a metric space 0 ⊆ such that the set of pointwise norm-attaining Lipschitz functions on 0 contains an isometric copy of 0. We also observe that there are countable metric spaces for which the set of pointwise normattaining Lipschitz functions contains an isometric copy of ∞, which is a result that does not hold for the set SNA() of strongly norm-attaining Lipschitz functions. Several new results on 0-embedding and 1-embedding into the set SNA() are presented as well. In particular, we show that if is a subset of an ℝ-tree containing all the branching points, then SNA() contains 0 isometrically. As a related result, we provide an example of metric space for which the set of norm-attaining functionals on the Lipschitz-free space over cannot contain an isometric copy of 0. Finally, we compare the concept of pointwise norm-attainment with the several different kinds of norm-attainment from the literature.