Point derivations for Lipschitz functions andClarke's generalized derivative

Vladimir Demyanov; Diethard Pallaschke

Applicationes Mathematicae (1997)

  • Volume: 24, Issue: 4, page 465-474
  • ISSN: 1233-7234

Abstract

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Clarke’s generalized derivative f 0 ( x , v ) is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed x ∈ X and fixed v ∈ E the function f 0 ( x , v ) is continuous and sublinear in f ∈ L i p ( X , d ) . It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.

How to cite

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Demyanov, Vladimir, and Pallaschke, Diethard. "Point derivations for Lipschitz functions andClarke's generalized derivative." Applicationes Mathematicae 24.4 (1997): 465-474. <http://eudml.org/doc/219186>.

@article{Demyanov1997,
abstract = {Clarke’s generalized derivative $f^0(x,v)$ is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed $x\in X$ and fixed $v\in E$ the function $f^0(x,v)$ is continuous and sublinear in $f\in Lip(X,d)$. It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.},
author = {Demyanov, Vladimir, Pallaschke, Diethard},
journal = {Applicationes Mathematicae},
keywords = {point derivations; generalized directional derivative; Lipschitz functions},
language = {eng},
number = {4},
pages = {465-474},
title = {Point derivations for Lipschitz functions andClarke's generalized derivative},
url = {http://eudml.org/doc/219186},
volume = {24},
year = {1997},
}

TY - JOUR
AU - Demyanov, Vladimir
AU - Pallaschke, Diethard
TI - Point derivations for Lipschitz functions andClarke's generalized derivative
JO - Applicationes Mathematicae
PY - 1997
VL - 24
IS - 4
SP - 465
EP - 474
AB - Clarke’s generalized derivative $f^0(x,v)$ is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed $x\in X$ and fixed $v\in E$ the function $f^0(x,v)$ is continuous and sublinear in $f\in Lip(X,d)$. It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.
LA - eng
KW - point derivations; generalized directional derivative; Lipschitz functions
UR - http://eudml.org/doc/219186
ER -

References

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  1. R. Arens and J. Eells, Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397-403. Zbl0073.39601
  2. F. H. Clarke, Optimization and Nonsmooth Analysis, CRM, Université de Montréal, 1989. Zbl0727.90045
  3. N. Dunford and J. T. Schwartz, Linear Operators: Part I, Interscience Publ. New York, 1957. 
  4. L. Hörmander, Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Ark. Mat. 3 (1954), 181-186. Zbl0064.10504
  5. D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-272. Zbl0121.10204
  6. I. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. Zbl0067.35101

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