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 c0$c_0$, 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 M$M$, there exists a metric space M0⊆M$M_0 \subseteq M$ such that the set of pointwise norm‐attaining Lipschitz functions on M0$M_0$ contains an isometric copy of c0$c_0$. We also observe that there are countable metric spaces M$M$ for which the set of pointwise norm‐attaining Lipschitz functions contains an isometric copy of ℓ∞$\ell _\infty$, which is a result that does not hold for the set SNA(M)$\operatorname{SNA}(M)$ of strongly norm‐attaining Lipschitz functions. Several new results on c0$c_0$‐embedding and ℓ1$\ell _1$‐embedding into the set SNA(M)$\operatorname{SNA}(M)$ are presented as well. In particular, we show that if M$M$ is a subset of an R$\mathbb {R}$‐tree containing all the branching points, then SNA(M)$\operatorname{SNA}(M)$ contains c0$c_0$ isometrically. As a related result, we provide an example of metric space M$M$ for which the set of norm‐attaining functionals on the Lipschitz‐free space over M$M$ cannot contain an isometric copy of c0$c_0$. Finally, we compare the concept of pointwise norm‐attainment with the several different kinds of norm‐attainment from the literature.