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

Abstract

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 $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-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, an earlier result of the second and third authors. 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.2000 Mathematical Subject Classification: 46G05, 46T20.

Smoothness, asymptotic smoothness and the Blum-Hanson property

“…The latter is one of the many moduli introduced by V. D. Milman in [39]. With the notation of [26], it is the functionρ X : R`ˆS X Ñ R`defined as follows:…”

On strong asymptotic uniform smoothness and convexity

“…We will use the same arguments that appear in Proposition 2.3. (3) in [16]. From the monotony of E follows that 1 2 ( x + ty − 1) ≤ 1 2 ( x + ty + x − ty ) − 1 whenever x ∈ H n ∩ S X and y ∈ H n ∩ S X for some n ∈ N. Thus,ρ E (t) ≤ 2ρ X (t).…”

“…In addition, ρ X is quantitatively related to δ * X by Young's duality. We refer the reader to [16] and the references therein for a detailed study of these properties. The related notions of nearly uniformly convex space (NUC for short) and nearly uniformly smooth (NUS for short) were introduced by Huff [15] and Prus [22].…”

Asymptotic uniform moduli and Kottman constant of Orlicz sequence spaces

“…Milman in [9] introduced two moduli for the study of an infinite-dimensional Banach space X. Johnson, Lindenstrauss, Preiss and Schechtman investigated these moduli in [7] and called them modulus of asymptotic uniform convexity, given for t > 0 by δ X (t) = inf The Banach space X is said to be asymptotically uniformly convex if δ X (t) > 0 for every 0 < t < 1, and asymptotically uniformly smooth if ρ X (t)/t → 0 as t → 0. For example, if X is a subspace of p , with 1 ≤ p < ∞, then, for every t…”