Norm attaining Lipschitz 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-Bollobás theorem is valid.

To appear in Banach Journal of Mathematical Analysis

References in corpus (1)

• The Bishop-Phelps-Bollobás moduli of a Banach space

Cited by in corpus (9)

• On strongly norm attaining Lipschitz maps
• The Bishop--Phelps--Bollobás property for Lipschitz maps
• Points of differentiability of the norm in Lipschitz-free spaces
• Residuality in the set of norm attaining operators between Banach spaces
• Emerging notions of norm attainment for Lipschitz maps between Banach spaces
• Convex integrals of molecules in Lipschitz-free spaces
• Normal functionals on Lipschitz spaces are weak${}^{\ast }$ continuous
• Some results on isometric composition operators on Lipschitz spaces
• Norm attaining Lipschitz maps toward vectors