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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Partial subdifferentials, derivates and Rademacher’s Theorem
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by D. N. Bessis and F. H. Clarke PDF
Trans. Amer. Math. Soc. 351 (1999), 2899-2926 Request permission


In this paper, we present new partial subdifferentiation formulas in nonsmooth analysis, based upon the study of two directional derivatives. Simple applications of these formulas include a new elementary proof of Rademacher’s Theorem in ${\mathbb R}^n$, as well as some results on Gâteaux and Fréchet differentiability for locally Lipschitz functions in a separable Hilbert space. Résumé. Dans cet article, nous présentons de nouvelles formules de sousdifférentiation partielle en analyse nonlisse, basées sur l’étude de deux dérivées directionnelles. Une simple application de ces formules nous permet d’obtenir une nouvelle preuve élémentaire du théorème de Rademacher dans ${\mathbb R}^{n}$, ainsi que certains résultats sur la différentiabilité Gâteaux ou Fréchet des fonctions localement Lipschitz sur un espace de Hilbert séparable.
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Additional Information
  • D. N. Bessis
  • Affiliation: Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London, United Kingdom, SW7 2AZ
  • Email:
  • F. H. Clarke
  • Affiliation: Mathématiques, Université de Lyon I, 69622 Villeurbanne, France, and Centre de Recherches Mathématiques, Université de Montréal, C. P. 6128, Succ. Centre-ville, Montréal, Québec, Canada, H3C 3J7
  • Email:
  • Received by editor(s): February 2, 1997
  • Published electronically: March 10, 1999
  • Additional Notes: We gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, and of le Fonds FCAR du Québec
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 2899-2926
  • MSC (1991): Primary 26E99; Secondary 46G05, 49J50, 58B10
  • DOI:
  • MathSciNet review: 1475676