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Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces
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Maleva, Olga and Preiss, David (2016) Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces. Transactions of the American Mathematical Society, 368 (7). pp. 4685-4730. doi:10.1090/tran/6480 ISSN 0002-9947.
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Official URL: http://dx.doi.org/10.1090/tran/6480
Abstract
Motivated by an attempt to find a general chain rule formula for differentiating the composition $ f\circ g$ of Lipschitz functions $ f$ and $ g$ that would be as close as possible to the standard formula $ (f\circ g)'(x) = f'(g(x))\circ g'(x)$, we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments. The idea behind these assignments is that the derivative of $ f$ at $ y$ is understood as defined only in the direction of a suitable ``tangent space'' $ U(f,y)$ (and so it exists at every point), but these tangent spaces are chosen in such a way that for any $ g$ they contain the range of $ g'(x)$ for almost every $ x$. Showing the existence of such assignments leads us to a detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions.
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Library of Congress Subject Headings (LCSH): | Lipschitz spaces, Banach spaces, Derivatives (Mathematics) | |||||||||
Journal or Publication Title: | Transactions of the American Mathematical Society | |||||||||
Publisher: | American Mathematical Society | |||||||||
ISSN: | 0002-9947 | |||||||||
Official Date: | 2016 | |||||||||
Dates: |
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Volume: | 368 | |||||||||
Number: | 7 | |||||||||
Page Range: | pp. 4685-4730 | |||||||||
DOI: | 10.1090/tran/6480 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Reuse Statement (publisher, data, author rights): | First published in Transactions of the American Mathematical Society in 368 (7). 2019, published by the American Mathematical Society. © Copyright 2015 American Mathematical Society | |||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||
Date of first compliant deposit: | 30 October 2019 | |||||||||
Date of first compliant Open Access: | 4 November 2019 | |||||||||
RIOXX Funder/Project Grant: |
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