We prove that for a given Banach space , the subset of norm-attaining Lipschitz functionals in is weakly dense but not strongly dense. Then we introduce a weaker concept of directional norm attainment and demonstrate that for a uniformly convex the set of directionally norm-attaining Lipschitz functionals is strongly dense in and, moreover, that an analogue of the Bishop–Phelps–Bollobás theorem is valid.
"Norm-attaining Lipschitz functionals." Banach J. Math. Anal. 10 (3) 621 - 637, July 2016. https://doi.org/10.1215/17358787-3639646