Generalized Gradients in Banach Space

Abstract

The calculus of generalized gradients is the best-known and most frequently invoked part of nonsmooth analysis. Unlike proximal calculus, it can be developed in an arbitrary Banach space X. In this chapter we make a fresh start in such a setting, but this time, in contrast to Chapter 1, we begin with functions and not sets. We present the basic results for the class of locally Lipschitz functions. Then the associated geometric concepts are introduced, including for the first time a look at tangency. In fact, we examine two notions of tangency; sets for which they coincide are termed regular and enjoy useful properties. We proceed to relate the generalized gradient to the constructs of the preceding chapter when X is a Hilbert space. Finally, we derive a useful limiting-gradient characterization when the underlying space is finite dimensional.