# zbMATH — the first resource for mathematics

Gateaux differentiable functions are somewhere Fréchet differentiable. (English) Zbl 0573.46024
Let G be a nonempty open subset of an Asplund space (i.e. such real Banach space, all of whose separable subspaces have a separable dual) and let $$f: G\to {\mathbb{R}}$$ be locally Lipschitz and Gâteaux differentiable at every point of G. Then the author has proved that f is Fréchet differentiable at uncountably many points of G. The nonseparable case is reduced to the separable one with the help of a general separable reduction theorem.
Reviewer: Duong Minh Duc

##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces 46B22 Radon-Nikodým, Kreĭn-Milman and related properties
Full Text:
##### References:
 [1] Aronszajn N.,Differentiability of Lipschitz mappings between Banach spaces, Studia Math.,57 (1976), 147–160. · Zbl 0342.46034 [2] Asplund E.,Frechet differentiability of convex functions, Acta Math.,121 (1968), 31–47. · Zbl 0162.17501 [3] Christensen J. P. R.,Measure theoretic zero sets in infinitely dimensional spaces and application to differentiability of Lipschitz mappings, Actes du Deuxieme Colloque d’Analyse Fonctionelle de Bordeaux, (Univ. de Bordeaux, 1973, no.2, 29–39. · Zbl 0302.43001 [4] Fitzpatrick S.,Metric projection and the differentiability of the distance functions, Bull. Austral. Math. Soc.,22 (1980), 291–312. · Zbl 0437.46012 [5] Leach F. B., Whitfield J. H. M.,Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc.,33 (1972), 120–126. · Zbl 0236.46051 [6] Kurzweil J.,On approximation in real Banach spaces, Studia Math.,14 (1954), 214–231. · Zbl 0064.10802 [7] Mankiewicz P.,On the differentiability of Lipschitz mappings in Frechet spaces, Studia Math.,45 (1973), 15–29. · Zbl 0219.46006 [8] Phelps R. R.,Gaussian null sets and differentiability of Lipschitz maps on Banach spaces, Pacific J. Math.,77 (1978), 523–531. · Zbl 0396.46041 [9] Preiss D.,Geteaux differentiable Lipschitz functions need not be Frechet differentiable on a residual set, Supplemento Rend. Circ. Mat. Palermo, serie II, no. 2 (1982), 217–222. · Zbl 0518.46032 [10] Stegall Ch.,The duality between Asplund spaces and spaces with the Radon-Nikodym property, Israel J. Math.,29 (1978), 408–412. · Zbl 0374.46015 [11] Stegall Ch.,The Radon-Nikodym property in conjugate Banach spaces II, Trans. Amer. Math. Soc.,264 (1981), no. 2, 507–519. · Zbl 0475.46016 [12] Zahorski Z.,Sur l’ensemble des points de non-derivabilite d’une fonction continue, Bull. Soc. Math. France,74 (1946), 147–178. · Zbl 0061.11302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.