Spaces of Lipscitz 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 $0 < \alpha < 1$, it is always true the space of Hölder continuous functions of class $\alpha$ is isomorphic to $\ell_\infty$. We show that, on the contrary, if $M$ is a compact convex subset of a Hilbert space this isomorphism holds if and only if $M$ is finite-dimensional. We also study the (related) problem of when a quotient map $Q:Y\rightarrow X$ between two Banach spaces admits a section which is uniformly continuous on the unit ball of $X$.