Jump to ContentJump to Main Navigation
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)$
Users without a subscription are not able to see the full content.

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

Show Summary Details
Page of

PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in HSO for personal use (for details see http://www.universitypressscholarship.com/page/privacy-policy).date: 20 April 2018

Fr ´Echet Differentiability of Vector-Valued Functions

Fr ´Echet Differentiability of Vector-Valued Functions

Chapter:
(p.262) Chapter Thirteen Fr ´Echet Differentiability of Vector-Valued Functions
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/9780691153551.003.0013

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

Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.