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

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 (Formula presented.), 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 (Formula presented.), there exists a metric space (Formula presented.) such that the set of pointwise norm-attaining Lipschitz functions on (Formula presented.) contains an isometric copy of (Formula presented.). We also observe that there are countable metric spaces (Formula presented.) for which the set of pointwise norm-attaining Lipschitz functions contains an isometric copy of (Formula presented.), which is a result that does not hold for the set (Formula presented.) of strongly norm-attaining Lipschitz functions. Several new results on (Formula presented.) -embedding and (Formula presented.) -embedding into the set (Formula presented.) are presented as well. In particular, we show that if (Formula presented.) is a subset of an (Formula presented.) -tree containing all the branching points, then (Formula presented.) contains (Formula presented.) isometrically. As a related result, we provide an example of metric space (Formula presented.) for which the set of norm-attaining functionals on the Lipschitz-free space over (Formula presented.) cannot contain an isometric copy of (Formula presented.). Finally, we compare the concept of pointwise norm-attainment with the several different kinds of norm-attainment from the literature.