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AN ALTERNATIVE APPROACH TO FRÉCHET DERIVATIVES

Part of: Functions of several variables Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  14 May 2020

SHANE ARORA Open the ORCID record for SHANE ARORA [Opens in a new window] ,
HAZEL BROWNE Open the ORCID record for HAZEL BROWNE [Opens in a new window]  and
DANIEL DANERS Open the ORCID record for DANIEL DANERS [Opens in a new window]
SHANE ARORA
Affiliation:
School of Mathematics & Statistics, University of Sydney, NSW 2006, Australia e-mail: saro0188@uni.sydney.edu.au
HAZEL BROWNE
Affiliation:
School of Mathematics & Statistics, University of Sydney, NSW 2006, Australia e-mail: hbro4811@uni.sydney.edu.au
DANIEL DANERS
Affiliation:
School of Mathematics & Statistics, University of Sydney, NSW 2006, Australia e-mail: daniel.daners@sydney.edu.au
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Abstract

We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.

Keywords

Fréchet derivativeSchwarz lemmaBanach fixed point theoremmean value theoremsCarathéodory derivative

MSC classification

Primary: 46G05: Derivatives
Secondary: 26B05: Continuity and differentiation questions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 2 , October 2021 , pp. 202 - 220
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by W. Moors

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