By Joram Lindenstrauss, W. B. Johnson

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**Example text**

K = [0, 1]), it is also known that Dωξ+1 = Dωξ . 13, ω α<ω1 Dα = D(K). Now following [43], deﬁne also the following transﬁnite analogues of B1/4(K), namely a transﬁnite descending family of Banach spaces (Vξ (K))ξ <ω1 , as follows. V1 (K) = B1/4 (K) and · 1 = · B1/4 (K) . A. Argyros et al. 17)}. Finally, for ξ a limit ordinal, Vξ (K) denotes the set of all f in α<ξ Vα (K) with f ξ =def supα<ξ f α < ∞. 6). 6, due to Farmaki ([43] and [44]). 8. 3), and let 1 ξ be a countable ordinal. (1) If f ∈ Vξ (K), there exists a convex block basis (yn ) of (xn ) which ξ -generates a spreading model equivalent to the summing basis.

10 involve reﬁnements of arguments in [16]. The basic concept of property (u) is introduced by Pełczy´nski in [106]; the terminology “DUC-sequences” was introduced in [117], just to focus attention on the structure of non-trivial weak-Cauchy sequences, as primary objects. 2, and x ∗∗ ∈ X∗∗ \ X is such that x ∗∗ |K ∈ DSC(K); that is, it is a difference of (possibly unbounded) semi-continuous functions on K. It is still the case that c0 embeds in X – cf. 1 of [66]. For a study of the class DSC(K), see [34].

5. The hypotheses imply that 1 does not embed in X, since c0 embeds in ( 1 )∗ and c0 is not weakly sequentially complete. A. Argyros et al. we may choose a bounded sequence in X with no weakly convergent subsequence. But this sequence in turn has a weak-Cauchy subsequence (xj ) by the 1 -theorem. 6, so since it is non-trivial weak-Cauchy, it has a convex block basis (fj ) equivalent to the summing basis, whence [fj ] is isomorphic to c0 , so c0 embeds in X. Moreover the same argument applies to any non-trivial weak-Cauchy sequence (xj ) in X; letting (fj ) be as above, (xj − fj ) is weakly null and (fj ) is DUC, so X has property (u).