In this chapter, we investigate spaces with M-bases and their relationship with several closely connected notions, such as projectional resolutions of identity, the separable complementation property, and projectional generators. Renorming theory also plays an important role here, as spaces with an M-basis admit an equivalent rotund norm, while spaces with a strong M-basis even have an LUR renorming thanks to the fundamental result of Troyanski.
The first section treats the existence issue; namely a space has a (strong) M-basis if it belongs to the P class. In the second section, it is shown that an M-basis can be linearly perturbed to become a bounded M-basis; this is a result of Plichko. The third and fourth sections treat various aspects of weakly Lindelöf determined (WLD) spaces, a class admitting many equivalent descriptions-in particular, as spaces whose dual ball is Corson and also as spaces admitting a weakly Lindelöf M-basis. The fifth section shows the impact of the additional axioms to ZFC on the structure of C(K) spaces, where K is a Corson compactum. The last two sections examine extensions of Mbases and quasicomplements. Among other things, it is shown that WLD spaces admit extensions of M-bases from subspaces, which implies that every subspace of a WLD space is quasicomplemented. The seventh section also contains Rosenthal’s theory of quasicomplements in \(\ell _\infty\) spaces.
KeywordsBanach Space Compact Space Equivalent Norm Separable Banach Space Asplund Space
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