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
pales@math.klte.hu
Vera Zeidan
Dept. of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.
zeidan@math.msu.edu
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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