On Fréchet differentiability of Lipschitz maps between Banach spaces

Joram Lindenstrauss¹ ; David Preiss²

¹ Einstein Institute of Mathematics, The Hebrew University of Jerusalem, 91905 Jerusalem, Israel
² Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

Annals of mathematics, Tome 157 (2003) no. 1, pp. 257-288.

Résumé

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space $X$ with separable dual has a common point of Fréchet differentiability. We show that the answer is positive for some infinite-dimensional $X$. Previously, even for collections consisting of two functions this has been known for finite-dimensional $X$ only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called $\Gamma$-null sets), whose definition involves both the notions of category and measure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Fréchet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces; no such results have been known previously.

Détail

DOI : 10.4007/annals.2003.157.257