Biorthogonal Systems in Nonseparable Spaces

The main theme of this chapter is the existence of biorthogonal systems in general nonseparable Banach spaces. An important role is played by the notion of long Schauder bases; the first section introduces this notion, which is a natural generalization of the usual Schauder basis. The first section also contains Plichko’s improvement of the natural “exhaustion” argument that yields the existence of a bounded total biorthogonal system in every Banach space. The second section presents Plichko’s characterization of spaces with a fundamental biorthogonal system as those spaces that admit a quotient of the same density with a long Schauder basis. In general, a total biorthogonal system as constructed in the first section has a cardinality that corresponds to the w*-density of the dual space. Thus, such a system may be countable for certain nonseparable spaces (most notably all subspaces of $\ell _\infty$). It is therefore a priori unclear if every nonseparable subspace of $\ell _\infty$ contains an uncountable biorthogonal system. The third section singles out some natural classes of spaces that are obtained “constructively” (representable spaces)— and hence are well-behaved in this respect, as shown by results of Godefroy and Talagrand. However, the general question of the existence of uncountable biorthogonal systems in every nonseparable space is undecidable in ZFC.