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 , we introduce and study a weaker notion of norm‐attainment for Lipschitz functions called the pointwise norm‐attainment. As a main result, we show that for every infinite metric space , there exists a metric space such that the set of pointwise norm‐attaining Lipschitz functions on contains an isometric copy of . We also observe that there are countable metric spaces for which the set of pointwise norm‐attaining Lipschitz functions contains an isometric copy of , which is a result that does not hold for the set of strongly norm‐attaining Lipschitz functions. Several new results on ‐embedding and ‐embedding into the set are presented as well. In particular, we show that if is a subset of an ‐tree containing all the branching points, then contains 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 . Finally, we compare the concept of pointwise norm‐attainment with the several different kinds of norm‐attainment from the literature.