Almost Frechet Differentiability of Lipschitz Mappings Between Infinite-Dimensional Banach Spaces

We give several sufficient conditions on a pair of Banach spacesX and Y under which each Lipschitz mapping from a domain inX to Y has, for every > 0, a point of -Fréchet differentiability.Most of these conditions are stated in terms of the moduli ofasymptotic smoothness and convexity, notions which have appearedin the literature under a variety of names. We prove, for example,that for > r > p 1, every Lipschitz mapping from a domainin an l_r-sum of finite-dimensional spaces into an l_p-sum offinite-dimensional spaces has, for every > 0, a point of-Fréchet differentiability, and that every Lipschitzmapping from an asymptotically uniformly smooth space to a finite-dimensionalspace has such points. The latter result improves, with a simplerproof, an earlier result of the second and third authors. Wealso survey some of the known results on the notions of asymptoticsmoothness and convexity, prove some new properties, and presentsome new proofs of existing results. 2000 Mathematical Subject Classification: 46G05, 46T20.