Diestel J. Geometry of Banach Spaces

Springer, 1975. — 289 p. — Series: Lecture Notes in Mathematics (Book 485) OCR
ISBN10: 3540074023 ISBN13: 978-3540074021These notes were the subject of lectures given at Kent State University during the 1973-74
academic year. At that time, it was already clear that the geometry of Banach spaces (in the form of convexity and smoothness type considerations) would play a central role in the theory of Radon-Nikodym differentiation for vector-valued measures. This was the object of the course: to acquaint my students (and, to a large extent, myself) with the geometry of Banach spaces. Naturally, the logical finish to the courses was a discussion of the Radon-Nikod m theorem viewed from a purely geometric perspective.The first chapter deals with the plenitude of support functionals to closed bounded convex subsets of a Banach space. Two results are focal: the Bishop-Phelps subreflexivity theorem and James' characterization of weak compactness. I feel that these are among the deepest results of modern functional analysis and have tried throughout the notes to apply them whenever possible. When the Deity allowed for theorems like these to be proved, He meant for them to be used! This chapter is closed with an application to operators attaining their norm which uses the theory to topological tensor products for its proof; this is the only excursion concerning prerequisites outside of elementary functional analysis and is a one-time affair. The principle purpose here is to highlight the severe restriction placed upon a Banach space (or pair of Banach spaces) that every operator attain its norm; it also is an interesting applicakion of James' theorem.Chapter Two deals with the basics of convexity and smoothness. It provides an excellent collection of applications of both principles set only on the geometric aspects of the Radon-Nikod m theorem; a much more comprehensive discussion (albeit from a different point of view) is contained in the forthcoming monograph, "Vector Measures" by J. J. Uhl, Jr. and myself.