DOI: 10.1515/ms-2017-0056 Math. Slovaca 67 (2017), No. 6, 1345–1358 RADEMACHER’S THEOREM IN BANACH SPACES WITHOUT RNP Donatella Bongiorno Dedicated to Professor Paolo de Lucia (Communicated by Anatolij Dvureˇcenskij ) ABSTRACT. We improve a Duda’s theorem concerning metric and w * -Gˆateaux differentiability of Lipschitz mappings, by replacing the σ-ideal A of Aronszajn null sets [ARONSZAJN, N.: Differentia- bility of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147–190], with the smaller σ-ideal ˜ A of Preiss-Zaj´ıˇcek null sets [PREISS, D.—ZAJ ´ I ˇ CEK, L.: Directional derivatives of Lipschitz functions, Israel J. Math. 125 (2001), 1–27]. We also prove the inclusion ˜ C o ⊂ ˜ A, where ˜ C o is the σ-ideal of Preiss null sets [PREISS, D.: Gˆ ateaux differentiability of cone-monotone and pointwise Lipschitz functions, Israel J. Math. 203 (2014), 501–534]. c 2017 Mathematical Institute Slovak Academy of Sciences Introduction The well known Rademacher’s theorem claims that a Lipschitz function from an open set G of R n into R m is Stoltz differentiable almost everywhere ([14], and [7: p. 216]). The extension of this theorem to infinite dimensional Banach spaces was largely investigated in the last years. Since in infinite dimensional Banach spaces there is no measure analogue to the Lebesgue measure (see, for example, A. V. Skorohod [15: p. 108]), and consequently the notion of “almost everywhere” cannot be defined by a measure, the extensions of Rademacher’s theorem obtained by Aronszajn [2], Christensen [4], Mankiewicz [10], Phelps [11], Preiss-Zaj´ıˇcek [13] and Preiss [12], are based on the definition of useful σ-ideals of null sets. The most direct extension of Stoltz differential is the notion of Fr´echet differential. Neverthe- less there exist some Lipschitz mappings between Hilbert spaces without a Fr´echet differential anywhere. So the notion of Gˆateaux differential was used to extend Rademacher’s theorem. Let X and Y be Banach spaces. A mapping f : X → Y is said to be Gˆateaux differentiable at x ∈ X provided that the limit df x (u) = lim t→0 f (x + tu) - f (x) t (1) exists for all u ∈ X r {0}, and df x (·) is a bounded linear operator. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 26B05; Secondary 49J50, 58C20, 28A15. K e y w o r d s: Lipschitz maps, Radon-Nikod´ ym property, metric Gˆateaux differentiability, w * -Gˆateaux differenti- ability. This work was supported by INDAM of Italy. 1345 Brought to you by | The University of Manchester Library Authenticated Download Date | 3/9/18 8:57 PM