Point derivations for Lipschitz functions andClarke's generalized derivative

• Autorzy: Vladimir F. Demyanov , Diethard Pallaschke
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

point derivations; generalized directional derivative; Lipschitz functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1996-1997
• Tom: 24
• Numer: 4
• Strony: 465-474

Daty

wydano

1997

otrzymano

1997-03-09

Twórcy

autor

Vladimir F. Demyanov

• Department of Applied Mathematics, St. Petersburg State University, 198904 Stary Peterhof, Russia

autor

Diethard Pallaschke

• Institut für Statistik und, Mathematische Wirtschaftstheorie, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany

Bibliografia

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