# On Fréchet differentiability of Lipschitz maps between Banach spaces | Annals of Mathematics

Abstract 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 Γ-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. Our main result states that a Lipschitz map between separable Banach spaces is Fréchet differentiable Γ-almost everywhere provided that it is regularly Gâteaux differentiable Γ-almost everywhere and the Gâteaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikodým property are Gâteaux differentiable Γ-almost everywhere. Moreover, Gâteaux differentiability implies regular Gâteaux differentiability with exception of another kind of negligible sets, so-called σ-porous sets. The answer to the question is therefore positive in every space in which every σ-porous set is Γ-null. We show that this holds for C(K) with K countable compact, the Tsirelson space and for all subspaces of c0, but that it fails for Hilbert spaces.

KEYWORDS

SHARE & LIKE

### 数学年刊（Annals of Mathematics）

0 Following 0 Fans 0 Projects 674 Articles

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Abstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Abstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Abstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Abstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Abstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Abstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge