ABSTRACT
The purpose of this chapter is to outline the basic aspects of the smooth and nonsmooth calculus in Banach spaces. Special emphasis is given on the nonsmooth theory, which started developing in the 1960’s, in order to provide a uniform viewpoint for the treatment of large classes of nonlinear extremal problems. The resulting subdifferential theories found also in many other applications and today are part of the so-called nonsmooth analysis, which is one of the most robust and interesting research areas of nonlinear functional analysis. In Section 4.1 we present the basics of the smooth calculus in Banach spaces.
We limit ourselves to the discussion of the Gaˆteaux and Fre´chet derivatives, which are the two most useful derivatives for vector valued functions. In Section 4.2 we consider convex functions defined on Banach spaces. We
discuss their continuity and differentiability properties. It turns out that a purely algebraic condition (convexity) has remarkable and powerful topological and differentiability implications. Also differentiability results bring us in contact with the Banach space theory and in particular with the so-called Asplund spaces which have the useful property that every separable subspace has a separable dual. We also show that every convex continuous function is locally Lipschitz. Locally Lipschitz functions between Banach spaces are the objects of investigation in Section 4.3. If the two Banach spaces are finite dimensional, then the locally Lipschitz function is differentiable almost everywhere (for the Lebesgue measure). Here we see how this can be generalized to the case where the two spaces are infinite dimensional. The main difficulty is to produce a suitable notion of negligible sets. This is done using the notion of Haar-null sets. We study them in detail and eventually prove an infinite dimensional version of the Rademacher theorem on the almost everywhere differentiability of locally Lipschitz functions. In Section 4.4 we pass to the nonsmooth part of this chapter. We examine
the duality and subdifferentiability properties of convex functions and the subdifferentiability properties of locally Lipschitz functions. At the end of the section, using the notion of bornology, we briefly consider some more subdifferentials of proper functions. In Section 4.5 we investigate integral functionals defined by convex or non-
convex normal integrands. We determine their duality and subdifferentiability
