By Raynaud M. (Ed), Shioda T. (Ed)

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2, (fn ) has a convex block basis equivalent to the summing basis. But if (gn ) is a convex block basis of (fn ) with (gn ) equivalent to the summing basis, then also gn → f pointwise and of course supk∈K |(gn − gn−1 )(k)| < ∞, so evidently f ∈ D(K). ), f ∈ / D(K). 10. ), a contradiction. 2 are mutually exclusive. 11(b). An important ingredient in the proof of this is an intrinsic characterization of functions in DBSC(K) (for K an arbitrary separable metric space), involving the transﬁnite oscillations of a given scalar valued functions on K.

7. Let K, (fn ), and g be as above, with g upper semi-continuous. Then given ε > 0, there exists a convex block basis (gn ) of (fn ) so that for all n and m, |gn − gm | + g ∞ < oscg f ∞ + ε. 5. Fix K a compact metric space, and for each countable inﬁnite ordinal α let Dα denote the family of all scalar valued bounded functions f with oscα f bounded. It can be seen that Dα is a Banach algebra under the norm f Dα = |f | + oscα f ∞ . One also has that for all 1 ξ and ωξ α < ωξ +1 , Dα = Dωξ ; in particular, this shows that rND (f ) = ωξ for some ξ , for any f .

These spaces are related to the famous Tsirelson space. The main intention is to present methods of producing Tsirelson and mixed Tsirelson norms. These are examples, or the frame for more advanced constructions, concerning the solutions of many important problems of Banach space theory. We also list properties of these spaces and we make a brief presentation of certain results for asymptotic p and Hereditarily Indecomposable spaces. 1. Compact families of ﬁnite subsets of N Throughout this section, M, N will denote families of ﬁnite subsets of N, which, in most of the cases, will be additionally assumed to be compact in the topology of pointwise con- Descriptive set theory and Banach spaces 1049 vergence.