Fr ´Echet Differentiability of Vector-Valued Functions

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
Fr ´Echet Differentiability of Vector-Valued Functions

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This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

Keywords: separable dual; Banach space; Fréchet smooth norm; modulus; Lipschitz map; Fréchet differentiability; Lipschitz function; Hilbert space; mean value estimate; regularity parameter

Chapter.  15173 words. 

Subjects: Mathematics

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