The delicate gradation of the different subclasses of the class of weakly Lindelöf determined (WLD) spaces is shown in this chapter through the optic of M-bases. It is known that WCG spaces can be characterized by the existence of a weakly compact M-basis, although not every M-basis in such a space is necessarily weakly compact (that it is so when using some more precise orthogonal structures will be investigated in Chapter 7). The subtle distinction between WCG spaces and subspaces of WCG spaces was brought to light when, in [Rose74], Rosenthal produced an example of the latter that was not WCG. Examples separating different important classes (WCG, their subspaces, Vašák, WLD) appear in the work of Rosenthal, Talagrand, Argyros, Mercourakis, and others. Here we reflect on these distinctions in terms of the existing M-bases using countable splittings of the systems.
More precisely, in this chapter we show that the class of WLD spaces and some of its subclasses, such as weakly compactly generated spaces and their subspaces, Vašák (i.e., WCD) spaces, and Hilbert generated spaces and their subspaces, can be characterized by the existence of M-bases with special covering properties. Moreover, in many cases, these properties of M-bases are then shared by all M-bases in the space. As applications of this, we present short proofs of the (uniform) Eberlein property for continuous images of (uniform) Eberlein spaces. Results of this nature were initially proved by using infinite combinatorial methods. We conclude this chapter with some results on spaces that are strongly generated by reflexive (resp. superreflexive) spaces. Using these results, we provide some short proofs of results on weakly compact sets in L1(μ) spaces where μ is a finite measure.
KeywordsBanach Space Bounded Linear Operator Dense Subset Equivalent Norm Unconditional Basis
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