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Journal of Convex Analysis 16 (2009), No. 2, 321--349
Copyright Heldermann Verlag 2009

The Core of the Infinite Dimensional Generalized Jacobian

Zsolt Páles
Institute of Mathematics, University of Debrecen, 4010 Debrecen, Pf. 12, Hungary

Vera Zeidan
Dept. of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.

Locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon-Nikodym property, a generalized core-Jacobian, Δ f(p) is introduced, and its fundamental properties are established. Primarily, it is shown that the β-closure of its convex hull is exactly the generalized Jacobian. Furthermore, the nonemptiness, the β-compactness, the β-upper semicontinuity, and even another representation are obtained. Connections with known notions are derived and chain rules are proved using key results developed. Therefore, the generalized core-Jacobian introduced in this paper is proved to enjoy all the properties that allow this set to be the nucleus of the generalized Jacobian.

Keywords: Generalized Jacobian.

MSC: 49J52, 49A52, 58C20

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