Generalized Jacobian for Functions with Infinite Dimensional Range and Domain

Abstract

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon-Nikodým property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set.

Co-Jacobian for Lipschitzian Maps

“…The space L(Y * , X * ) has the property that X * has X as a predual, and hence it falls in the setting of [19]. Therefore, we introduce in L(Y * , X * ) a topology in which the norm-closed unit ball of L(Y * , X * ) is compact.…”

“…For the case when Y is an infinite dimensional dual space with the Radon-Nikodým property, an involved generalization of the above construction was elaborated in [19] which led to an extension in this setting of the generalized Jacobian ∂ f (·) defined by (2.7). The underlying technique employed in [19] is the generalization of Rademacher's differentiability theorem proven in [2] and [1].…”

“…The underlying technique employed in [19] is the generalization of Rademacher's differentiability theorem proven in [2] and [1]. This notion will also be compared to the co-Jacobian in Section 9.…”

“…Recently, in a series of papers [19][20][21], and [22] the authors have developed a notion of a generalized Jacobian for Lipschitzian functions defined on any normed space X. In the papers [20] and [21] the generalized Jacobian for the case when the range space Y is finite dimensional was constructed so that it encompasses as special cases both Clarke's generalized Jacobian and gradient.…”

“…In the papers [20] and [21] the generalized Jacobian for the case when the range space Y is finite dimensional was constructed so that it encompasses as special cases both Clarke's generalized Jacobian and gradient. In [19] however, this notion was extended to the case when the range Y is a dual space with the Radon-Nikodým property. It is worth mentioning that the Radon-Nikodým property of Y guarantees the applicability of the generalized Rademacher differentiability theorem, while the range being a dual space ensures that the norm-closed unit ball of the space L(X, Y) is compact in a certain topology.…”