Chapter 50 - Non-smooth analysis, optimisation theory and Banach space theory

Abstract

This chapter discusses about nonsmooth analysis, optimization theory, and Banach space theory. A Banach space X is called a weak Asplund space [Gâteaux differentiability space] if each continuous convex function defined on it is Gâteaux differentiable at the points of a residual subset (that is, a subset that contains the intersection of countably many dense open subsets of X) [dense subset] of its domain. For a Banach space (X, ∥ ·∥ ), with closed unit ball BX, the Bishop–Phelps set is the set of all linear functionals in the dual X* that attain their maximum value over BX; that is, the set {x*∈X* : x*(x) = ∥x*∥ for some x ∈BX}. The Bishop–Phelps Theorem says that the Bishop–Phelps set is always dense in X*. A Banach space X has the attainable approximation property (AAP) if the set of support functionals for any closed bounded convex subset W ⊆X is norm dense in X*. The concepts related to the Bishop–Phelps problem and the complex Bishop–Phelps property are also discussed. Concepts of biorthogonal sequences and support points are also elaborated.