Norm-attaining Lipschitz functionals

Abstract

We prove that for a given Banach space $X$, the subset of norm-attaining Lipschitz functionals in ${Lip}_0\left(X\right)$ is weakly dense but not strongly dense. Then we introduce a weaker concept of directional norm attainment and demonstrate that for a uniformly convex $X$ the set of directionally norm-attaining Lipschitz functionals is strongly dense in ${Lip}_0\left(X\right)$ and, moreover, that an analogue of the Bishop–Phelps–Bollobás theorem is valid.