Nonlinear structure of some classical quasi-Banach spaces and F-spaces ✩

We investigate the Lipschitz structure of p and Lp for 0 < p < 1 as quasi-Banach spaces and as metric spaces (with the metric induced by the p-norm) and show that they are not Lipschitz isomorphic. We prove that the F-space L0 is not uniformly homeomorphic to any other Lp space, that the Lp spaces for 0 < p <1 embed isometrically into one another, and reduce the problem of the uniform equivalence amongst Lp spaces to their Lipschitz equivalence as metric spaces. © 2007 Elsevier Inc. All rights reserved.