Subspaces of $ c_0 ({\Bbb N}) $ and Lipschitz isomorphisms

Abstract

We show that the class of subspaces of \( c_0 ({\Bbb N}) \) is stable under Lipschitz isomorphisms. The main corollary is that any Banach space which is Lipschitz-isomorphic to \( c_0 ({\Bbb N}) \) is linearly isomorphic to \( c_0 ({\Bbb N}) \). The proof relies in part on an isomorphic characterization of subspaces of \( c_0 ({\Bbb N}) \) as separable spaces having an equivalent norm such that the weak-star and norm topologies quantitatively agree on the dual unit sphere. Estimates on the Banach—Mazur distances are provided when the Lipschitz constants of the isomorphisms are small. The quite different non-separable theory is also investigated.