Norm attaining Lipschitz functionals

TL;DR

This paper investigates the density properties of norm attaining Lipschitz functionals on Banach spaces, introducing a weaker notion called directional norm attainment, and establishes density results and an analogue of the Bishop-Phelps-Bollobás theorem.

Contribution

It proves weak density of norm attaining Lipschitz functionals and strong density of directionally norm attaining functionals in uniformly convex spaces, along with a Bishop-Phelps-Bollobás type result.

Findings

Norm attaining Lipschitz functionals are weakly dense but not strongly dense.

Directionally norm attaining functionals are strongly dense in uniformly convex spaces.

An analogue of the Bishop-Phelps-Bollobás theorem holds for directionally norm attaining functionals.

Abstract

We prove that for a given Banach space $X$, the subset of norm attaining Lipschitz functionals in $\mathrm{Lip}_0(X)$ 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 $\mathrm{Lip}_0(X)$ and, moreover, that an analogue of the Bishop-Phelps-Bollob\'as theorem is valid.