Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Fréchet vs. Gâteaux differentiability of Lipschitzian functions

Authors: Maria Gieraltowska-Kedzierska and F. S. Van Vleck
Journal: Proc. Amer. Math. Soc. 114 (1992), 905-907
MSC: Primary 26E15; Secondary 26A16, 46G05, 49J50
MathSciNet review: 1010805
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Examples have been given of Lipschitzian functions that are Gâteaux-differentiable everywhere, but nowhere Fréchet-differentiable. One such example has been reported, mistakenly, in several papers as having domain in $ {L^2}([0,\pi ])$, when it should have been $ {L^1}([0,\pi ])$. We discuss this example.

References [Enhancements On Off] (What's this?)

  • [1] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147-190. MR 0425608 (54:13562)
  • [2] J.-P. Aubin and I. Ekeland, Applied nonlinear analysis, Wiley, New York, 1984. MR 749753 (87a:58002)
  • [3] V. I. Averbukh and O. G. Smolyanov, The theory of differentiation in linear topological spaces, Russian Math. Surveys 22 (1967), 201-258.
  • [4] J. P. R. Christensen, Topological and Borel structure, North-Holland Math. Stud., vol. 10, 1974.
  • [5] M. Z. Nashed, Differentiability and related properties of nonlinear operators: Some aspects of differentials in nonlinear functional analysis, Nonlinear Functional Analysis (L. B. Rall, ed.), Academic Press, New York, 1971. MR 0276840 (43:2580)
  • [6] R. R. Phelps, A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976-983. MR 0123172 (23:A501)
  • [7] M. Sova, Conditions for differentiability in linear topological spaces, Czechoslovak Math. J. 16 (1966), 339-362.(Russian) MR 0198197 (33:6356)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26E15, 26A16, 46G05, 49J50

Retrieve articles in all journals with MSC: 26E15, 26A16, 46G05, 49J50

Additional Information

Keywords: Fréchet differentiability, Gâteaux differentiability
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society