More Applications

In this chapter, we discuss some further applications of techniques of orthogonalization in the geometry of Banach spaces. In particular, this chapter begins by showing the connection between a form of one-sided biorthogonal systems and the existence of support sets in nonseparable Banach spaces. Set theory once again plays an important role because, building on some previous work of Rolewicz, Borwein, Kutzarova, Lazar, Bell, Ginsburg, Todorčević, and others, recent results of Todorčević and Koszmider demonstrate that the existence of support sets in every nonseparable Banach space is undecidable in ZFC. The second section highlights some work of Granero, Jiménez-Sevilla, Moreno, Montesinos, Plichko, and others on the study of nonseparable Banach spaces that do not admit uncountable biorthogonal systems (the existence of such spaces relies on the use of additional axioms such as ♣); these results include characterizations of spaces in which every dual ball is weak*-separable, as well as an improvement of Sersouri’s result showing that such spaces must only contain countable ω-independent families.

The latter part of the chapter presents several applications of various types of biorthogonal systems. In particular, it is shown that fundamental biorthogonal systems (and even weaker systems) have applications in the study of norm-attaining operators as originated by Lindenstrauss. The attention then shifts to the Mazur intersection property, where, among other things, an application of biorthogonal systems to renorming Banach spaces with the Mazur intersection property is presented. The chapter concludes by showing that every Banach space can be renormed to have only trivial isometries; the proof of this relies on the fact that every Banach space has a total biorthogonal system.