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Embedding into Banach spaces with finite dimensional decompositions. (English) Zbl 1118.46018

This survey paper is devoted to an exposition of the main results of the authors’ papers [Trans.Am.Math.Soc.354, No.10, 4085–4108 (2002; Zbl 1023.46014) and Math.Ann.335, No.4, 901–916 (2006; Zbl 1108.46007)] (see these reviews for details) in a unified manner and with complete proofs.
Key to the arguments is combinatorial work discussed in Section 2 and applied to characterise spaces with separable duals having Szlenk index \(\omega\) in Section 3. Section 4 discusses embeddability into spaces having finite-dimensional decompositions with certain estimates; and Section 5 contains the solution of a problem raised by J. Bourgain to the effect that there is a separable reflexive space that is universal for the class of all separable superreflexive spaces.
The authors also give a proof of N.Kalton’s \(c_0\)-theorem [Q.J.Math.52, No.3, 313–328 (2001; Zbl 1016.46012)] and mention as yet unpublished joint work with A.Zsák.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
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