Almost Fréchet differentiability of Lipschitz mappings between infinite dimensional Banach spaces

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every ǫ > 0, a point of ǫ-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which appeared in the literature under a variety of names. We prove, for example, that for ∞ > r > p ≥ 1, every Lipschitz mapping from a domain in an lr sum of finite dimensional spaces into an lp sum of finite dimensional spaces has, for every ǫ > 0, a point of ǫ-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite dimensional space has such points. The latter result improves, with a simpler proof, the result of [16]. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results. Supported in part by NSF DMS-9900185, Texas Advanced Research Program 010366-163, and the U.S.-Israel Binational Science Foundation. Supported in part by the U.S.-Israel Binational Science Foundation. MR subject classification: 46G05, 46T20.