Closed linear spaces consisting of strongly norm attaining Lipschitz functionals

Abstract

AbstractGiven a pointed metric space M, we study when there exist n-dimensional linear subspaces of $$\mathrm {Lip}_0(M)$$ Lip 0 ( M ) consisting of strongly norm-attaining Lipschitz functionals, for $$n\in {\mathbb {N}}$$ n ∈ 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 for a metric space M in order to such a subspace Y exist. We also show that if the metric space M is $$\sigma $$ σ -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.