By Joram Lindenstrauss, W. B. Johnson

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**Extra resources for Handbook of the Geometry of Banach Spaces : Volume 2**

**Example text**

4, is due to Bourgain [24]. Actually, several basic notions in Chapter I ﬁrst appear in [24]. 5) was discovered by Argyros and Kanellopoulos “in between” drafts of the present paper. 1 that if a separable Banach space X contains sequences ξ -generating a spreading model equivalent to a given 1-spreading basis (ej ) for every countable ordinal ξ , then E embeds in X, where E is the Banach space with basis (ej ). Now the Bourgain 1 -index theorem investigates sequences ξ -generating the 1 -basis, while the c0 -index theorem studies sequences ξ -generating the summing basis.

Does there exist a separable X satisfying the hypotheses of Problem 2 such that X is cotype 2? For the next problem, we deﬁne (following [64]), the arbitrary intrinsic Baire-classes of the double dual of X, as follows. Set X0∗∗ = X. Suppose β > 0 is a countable ordinal and Xα∗∗ is deﬁned for all α < β. Let Xβ = {x ∗∗ ∈ X∗∗ : there exists a sequence (xn∗∗ ) of elements of α<β Xα∗∗ with xn∗∗ → ∗∗ . x ∗∗ ω∗ }. 2(a) yields that X1∗∗ = XB 1 ∗∗ = X ∗∗ for all α < ω . Is X universal? P ROBLEM 4. Assume that Xα+1 1 α Of course the theorem in [102] yields that X2∗∗ = X1∗∗ implies 1 embeds in X.

And moreover (fj ) is weak-Cauchy with fj → x ∗∗ ω∗ . By part (a), there exists also a weakCauchy sequence, (xj ) in X with xj → x ∗∗ ω∗ . But then fj − xj → 0 weakly and (fj ) is / X). 4) holds. We recall ﬁnally the following concept. 12 ([107]). A Banach space X has property (u) provided for any weakCauchy sequence (xj ) in X, there exists a DUC sequence (yj ) in X with xj − yj → 0 weakly. 10. 13. Let X be a given Banach space. The following are equivalent. (1) X has property (u). (2) Every non-trivial weak-Cauchy sequence in X has a convex block basis equivalent to the summing basis.