Spaces of Lipschitz and Hölder functions and their applications

We study the structure of Lipschitz and Hölder-type spaces and their preduals on general metric spaces, and give applications to the uniform structure of Banach spaces. In particular we resolve a problem of Weaver who asks whether if M is a compact metric space and 0 < α < 1, it is always true the space of Hölder continuous functions of class α is isomorphic to `∞. We show that, on the contrary, if M is a compact convex subset of a Hilbert space this isomorphism holds if and only ifM is finite-dimensional. We also study the (related) problem of when a quotient map Q:Y → X between two Banach spaces admits a section which is uniformly continuous on the unit ball of X.