Two unexpected examples concerning differentiability of Lipschitz functions on Banach spaces

Preiss1 and Jaroslav Tǐser2 In this note we present two examples illustrating some surprising rela-tions between the known existence theorems concerning two main concepts of differentiability of Lipschitz functions between Banach spaces. Recall that these concepts are: Gâteaux derivative of a mapping ϕ: X 7 → Y at x ∈ X, which is defined as a continuous linear map ϕ′(x) : X 7 → Y verifying 〈ϕ′(x), u 〉 = lim t→0 ϕ(x+ tu) − ϕ(x) t for every u ∈ X, and Fréchet derivative which, in addition, requests that the above limit be uniform for ‖u ‖ ≤ 1. The examples we give point out that our understanding of differentiability properties of (real-valued) Lipschitz functions on Banach spaces is far from being complete. Since the motivation behind each of them is quite different, we will explain it in detail at the beginning of each of the two sections. Here we will confine ourselves to basic motivational remarks in the case of an infinitely dimensional separable Hilbert space H. Our first example comes from the observation that the proof of existence of a point of Fréchet differentiability of a Lipschitz function f: H 7 → R in [8] appears to be unnatural in the sense that it doesn’t use the existing strong results about Gâteaux differentiability: Since we know that f is Gâteaux differentiable on a set E ⊂ H whose complement is “negligible”, the natura