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 wether 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 l∞. We show that, on the contrary, if M is a compact convex subset of a Hilbert space this isomorphism holds if and only if...

Given a compact manifold $N^n$, an integer $k\in {\mathbb{N}}_{*}$ and an exponent $1\le p<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k,p}(Q^m;N^n)$ when the homotopy group ${\pi }_{\lfloor kp\rfloor }\left(N^n\right)$ of order $\lfloor kp\rfloor$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m-\lfloor kp\rfloor -1$, without any restriction on the homotopy group of $N^n$.