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Norm attaining Lipschitz functionals

Research paper by Vladimir Kadets, Miguel Martin, Mariia Soloviova

Indexed on: 28 Jan '16Published on: 28 Jan '16Published in: Mathematics - Functional Analysis



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.