Compact reduction in Lipschitz-free spaces

Abstract

[EN] We prove a general principle satisfied by weakly precompact sets of Lip-schitz-free spaces. By this principle, certain infinite-dimensional phenomena in Lipschitzfree 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: F(X) is weakly sequentially complete for every superreflexive Banach space X, and F(M) has the Schur property and the approximation property for every scattered complete metric space M.