Fréchet vs. Gâteaux differentiability of Lipschitzian functions

Gieraltowska-Kedzierska, Maria; Van Vleck, F. S.

doi:10.1090/S0002-9939-1992-1010805-X

Fréchet vs. Gâteaux differentiability of Lipschitzian functions
HTML articles powered by AMS MathViewer

by Maria Gieraltowska-Kedzierska and F. S. Van Vleck PDF

Proc. Amer. Math. Soc. 114 (1992), 905-907 Request permission

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

N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147–190. MR 425608, DOI 10.4064/sm-57-2-147-190
Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753

The theory of differentiation in linear topological spaces

22

Topological and Borel structure

M. Z. Nashed, Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 103–309. MR 0276840
R. R. Phelps, A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960), 976–983. MR 123172, DOI 10.1090/S0002-9939-1960-0123172-X
Miroslav Sova, Conditions for differentialbility in linear topological spaces, Czechoslovak Math. J. 16(91) (1966), 339–362 (Russian, with English summary). MR 198197, DOI 10.21136/CMJ.1966.100737