Closed linear spaces consisting of strongly norm attaining Lipschitz mappings

Abstract

Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of ${\mathrm{Lip}}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in \mathbb{N}$. We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces $Y$, as well as the inverse question, that is, the possible sizes of the metric space $M$ given that such a subspace $Y$ exists. We also show that if the metric space $M$ is $\sigma$-precompact, then the aforementioned subspaces $Y$ need to be always separable and isomorphically polyhedral, and we show that for spaces containing $[0,1]$ isometrically, they can be infinite-dimensional.

arXiv commentSome of the results (mostly Section 2) have been improved since the first version of the preprint (arXiv:2202.06855v1) thanks to a comment that Mikhail Ostrovskii kindly made to us. The new version has 12 pages