Literature cited
D. Preiss, “Gâteaux-differentiable functions are somewhere Fréchet differentiable,” Rend. Circ. Mat. Palermo (2),33, No. 1, 122–133 (1984).
D. Preiss, “Gâteaux-differentiable Lipschitz functions need not be Fréchet differentiable on a residual set,” Supplemento Rend. Circ. Mat. Palermo (2), No. 2, 217–222 (1982).
V. I. Averbukh and O. G. Smolyanov, “Different definitions of the derivative in linear topological spaces,” Usp. Mat. Nauk,23, No. 4, 67–116 (1968).
V. I. Averbukh, O. G. Smolyanov, and S. V. Fomin, “Generalized functions and differential equations in linear spaces,” Trudy MMO,24, 132–174 (1971).
J. L. Kelley, General Topology, Van Nostrand, Princeton, N.J. (1955).
C. Kuratowski, Topologie, I, Warsaw (1948).
G. Lebourg, “Generic differentiability of Lipschitzian functions,” Trans. Am. Math. Soc.,256, 125–144 (1979).
M. Sova, “Differentiability conditions in linear topologial spaces,” Chekhoslov. Mat. Zh.,16, No. 3, 339–362 (1966).
N. Aronszajn, “Differentiability of Lipschitzian mappings between Banach spaces,” Stud. Math.,57, No. 2, 147–190 (1976).
J. Diestel, Geometry of Banach Spaces — Selected Topics, Springer, Berlin (1975).
A. M. Bruckner, Differentiation of Real Functions, Springer, Heidelberg (1978).
B. J. Pettis, “On the Radon-Nikodym theorem,” Lect. Notes Mat.,644, 340–355 (1978).
R. R. Phelps, “Gaussian null sets and differentiability of Lipschitz maps on Banach spaces,” Pac. J. Math.,77, No. 3, 523–531 (1978).
V. I. Bogachev, “Negligible sets and differentiable measures in Banach spaces,” Vestn. Mosk. Gos. Univ. Ser. 1. Mat., Mekh., No. 3, 47–52 (1982).
V. Goodman, “Quasidifferentiable functions on Banach spaces,” Proc. Am. Math. Soc.,30, No. 2, 367–370 (1971).
L. Gross, “Potential theory on Hilbert space,” J. Func. Anal.,1, 123–181 (1967).
R. Bonic and J. Frampton, “Differentiable functions on certain Banach spaces,” Bull. Am. Math. Soc.,71, No. 2, 393–395 (1965).
A. S. Nemirovskii and S. M. Semenov, “On polynomial approximation of functions on Hilbert space,” Mat. Sb.,92, No. 2, 257–281 (1973).
J. Kurzweil, “On approximation in real Banach spaces,” Stud. Math.,14, No. 2, 214–231 (1954).
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Translated from Matematicheskie Zametki, Vol. 44, No. 5, pp. 567–583, November, 1988.
The authors would like to thank O. G. Smolyanov for his attention to the paper and S. V. Konyagin for his useful comments.
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Bogachev, V.I., Shkarin, S.A. Differentiable and Lipschitzian mappings of Banach spaces. Mathematical Notes of the Academy of Sciences of the USSR 44, 790–798 (1988). https://doi.org/10.1007/BF01158417
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DOI: https://doi.org/10.1007/BF01158417