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 M 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.
@article{urn:eudml:doc:44337,
title = {Spaces of Lipschitz and H\"older functions and their applications.},
journal = {Collectanea Mathematica},
volume = {55},
year = {2004},
pages = {171-217},
zbl = {1069.46004},
mrnumber = {MR2068975},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44337}
}
Kalton, Nigel J. Spaces of Lipschitz and Hölder functions and their applications.. Collectanea Mathematica, Tome 55 (2004) pp. 171-217. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44337/