Infinite dimensional generalized Jacobian: Properties and calculus rules

The extension to infinite dimensional domains of Clarke's generalized Jacobian is the focus of this paper. First, a generalization of a Fabian–Preiss theorem to the infinite dimensional setting is obtained. As a consequence, a new formula relating the Clarke's generalized Jacobians corresponding to finite dimensional spaces K, L with K⊆L is established. Furthermore, in the infinite dimensional case, basic properties pertaining the generalized Jacobian are developed and then an identification of this set-valued map is produced. Applications of these results in the form of chain rules including sum and product rules, and a computational formula for continuous selections are derived.