COARSE AND UNIFORM EMBEDDINGS INTO REFLEXIVE SPACES

Abstract

Answering an old problem in nonlinear theory, we show that c0 cannot be coarsely or uniformly embedded into a reflexive Banach space, but that any stable metric space can be coarsely and uniformly embedded into a reflexive space. We also show that certain quasi-reflexive spaces (such as the James space) also cannot be coarsely embedded into a reflexive space and that the unit ball of these spaces cannot be uniformly embedded into a reflexive space. We give a necessary condition for a metric space to be coarsely or uniformly embeddable in a uniformly convex space.