Chapter

Ε‎-Fr ´Echet Differentiability

Joram Lindenstrauss, David Preiss and Tiˇser Jaroslav

in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Published by Princeton University Press

Published in print February 2012 | ISBN: 9780691153551
Published online October 2017 | e-ISBN: 9781400842698
Ε‎-Fr ´Echet Differentiability

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This chapter treats results on ε‎-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives. The chapter first presents a simple proof of an almost differentiability result for Lipschitz functions in asymptotically uniformly smooth spaces before discussing the notion of asymptotic uniform smoothness. It then proves that in an asymptotically smooth Banach space X, any finite set of real-valued Lipschitz functions on X has, for every ε‎ > 0, a common point of ε‎-Fréchet differentiability.

Keywords: asymptotically smooth space; ε‎-Fréchet differentiability; Lipschitz function; Frechet differentiability; Fréchet derivative; Gâteaux derivative; asymptotic uniform smoothness; Banach space

Chapter.  9636 words. 

Subjects: Mathematics

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