Weak Topologies and Banach Spaces

Abstract

An indispensable tool in the study of deeper structural properties of a Banach space X is its weak topology, i.e., the topology on X of the pointwise convergence on elements of the dual space X * , or the weak * topology on X * , i.e., the topology on X * of the pointwise convergence on elements of X. The topology on X * of the uniform convergence on the family of all convex balanced and weakly compact subsets of X plays also an important role. All those topologies can be efficiently studied in the general framework of topological vector spaces. This allows the use of Tychonoff's compactness theorem for weak * topologies in duals to Banach spaces and the results of Banach-Dieudonné type. We also study extreme points, the Choquet representation theorem, properties of James boundaries , and characterizations of weakly compact sets. We briefly discuss nonlocally convex spaces and the space of distributions. We then study the third fundamental principle of Functional Analysis-the Hahn-Banach theorem and the Banach open mapping principle being the subject of the previous chapter-, namely the Banach-Steinhaus uniform boundedness principle. Finally we introduce and study reflexive Banach spaces.