Multifunctional and functional analytic techniques in nonsmooth analysis

Abstract

These lectures center on the structure of real—valued Lipschitz functions, and their generalized derivatives on Banach spaces. We pay some attention to the role of measure and category and will try to illustrate a number of different techniques. These published notes are much more detailed and comprehensive than the lectures as given. Much of this development is based on recent joint work with Warren Moors (Wellington) and others. The exposition will be organized around the following interwoven themes. • Set—valued analysis. Minimal upper semi-continuous multifunctions (‘uscos and cuscos’) and related topological tools. Selections and single-valuedness. Relations to differentiability. • Measure and category as competing notions of smallness: Haar null sets and generic sets in Banach spaces. Other concepts of prevalence. • ‘Utility grade’ renorming theory and its application to the study of viscosity sub-derivatives and (partially) smooth variational principles. “Fuzzy” calculus and some equivalent reformulations. • The structure of Lipschitz functions. Especially the calculus of essentially smooth Lipschitz functions and the vector-lattice algebra they generate. Chain rules, and questions of integrability and representability. • Applications to and examples of distance functions. Minimality of distance functions and proximal normal formulae revisited. More general perturbation functions. • Convex functions and related sequences in Banach space. How properties of given Banach spaces are reflected in the convex functions they support. Conjugates and subdifferentials of eigenvalue functions.