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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)$
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Joram Lindenstrauss, David Preiss, and Jaroslav Tier

Print publication date: 2012

Print ISBN-13: 9780691153551

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153551.001.0001

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Porosity and ε‎-Fr échet differentiability

Porosity and ε‎-Fr échet differentiability

(p.202) Chapter Eleven Porosity and ε‎-Fr échet differentiability
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Joram Lindenstrauss

David Preiss

Tiˇser Jaroslav

Princeton University Press

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.

Keywords:   porous sets, irregularity point, Lipschitz function, ε‎-Fréchet differentiability, slice, Gâteaux derivative, Lipschitz map

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