Compact reduction in Lipschitz-free spaces
Volume 260 / 2021
Studia Mathematica 260 (2021), 341-359
MSC: Primary 46B20; Secondary 54E50.
DOI: 10.4064/sm200925-18-1
Published online: 19 April 2021
Abstract
We prove a general principle satisfied by weakly precompact sets of Lips\-chitz-free spaces. By this principle, certain infinite-dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.
