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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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Differentiability of Lipschitz Maps on Hilbert Spaces

Differentiability of Lipschitz Maps on Hilbert Spaces

Chapter:
(p.392) Chapter Sixteen Differentiability of Lipschitz Maps on Hilbert Spaces
Source:
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)
Author(s):

Joram Lindenstrauss

David Preiss

Tiˇser Jaroslav

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153551.003.0016

This chapter presents a separate, essentially self-contained, nonvariational proof of existence of points of Fréchet differentiability of R²-valued Lipschitz maps on Hilbert spaces. It begins with the theorem stating that every Lipschitz map of a Hilbert space to a two-dimensional space has points of Fréchet differentiability. This is followed by a lemma, which is stated in an arbitrary Hilbert space but whose validity in the general case follows from its three-dimensional version. The chapter then explains the proof of the theorem and of the lemma stated above. In particular, it considers two cases, one corresponding to irregular behavior and the other to regular behavior.

Keywords:   regular behavior, Fréchet differentiability, Lipschitz map, Hilbert space, two-dimensional space, irregular behavior

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