Weak Topologies and Banach Spaces

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.