Geometry of Banach Spaces: Selected Topics
Geometry of Banach Spaces: Selected Topics
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DOI：
10.1007/bfb0082079
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发表时间：
1975
期刊：
影响因子：
--
通讯作者：
J. Diestel
中科院分区：
文献类型：
--
作者：
J. Diestel
These 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. courses was a discussion of the Radon-Nikodým theorem viewed from a Naturally, the logical finish to the purely geometric perspective. Some words about the organization of the notes. 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 application of James' theorem.
