The Core of the Infinite Dimensional Generalized Jacobian

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.