Linear structures of norm-attaining Lipschitz functions and their complements

We solve two main questions on linear structures of (non-)norm-attaining Lipschitz functions. First, we show that for every infinite metric space M , the set consisting of Lipschitz functions on M which do not strongly attain their norm and the zero function contains an isometric copy of ℓ_∞, and moreover, those functions can be chosen not to attain their norm as functionals on the Lipschitz-free space over M . Second, we prove that for every infinite metric space M , neither the set of strongly norm-attaining Lipschitz functions on M nor the union of its complement with zero is ever a linear space. Furthermore, we observe that the set consisting of Lipschitz functions which cannot be approximated by strongly norm-attaining ones and the zero element contains ℓ_∞ isometrically in all the known cases. Some natural observations and spaceability results are also investigated for Lipschitz functions that attain their norm in one way but do not in another.