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On Baire-1/4 functions and spreading models

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  26 February 2010

Vassiliki Farmaki
Vassiliki Farmaki
Affiliation:
Department of Mathematics, University of Athens, Panepistemiopolis, llissia, Greece.
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Abstract

We prove a characterization of functions in B1/4(K)\C(K), where K is a compact metric space in terms of c0-spreading models, answering a Problem of R. Haydon, E. Odell and H. Rosenthal. Beginning with B1/4(K) we define a decreasing family (Vξ(K),║ · ║ξ)1≤ξ<ω1 of Banach spaces whose intersection is DBSC(K) and we prove an analogous stronger property for the functions in Vξ(K)\C(K). Defining the s-spreading model-index, we classify B1/4;(K) and we prove that s-SM[F]>ξ for every F∈ Vξ(K). Also we classify the separable Banach spaces by defining the c0-SM-index which measures the degree to which they have sequences with extending spreading models equivalent to the usual basis of c0. We give examples of Baire-1 functions and reflexive spaces with arbitrary large indices.

MSC classification

Secondary: 46E99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 41 , Issue 2 , December 1994 , pp. 251 - 265
Copyright
Copyright © University College London 1994

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References

1.Alspach, D. and Argyros, S.. Complexity of weakly null sequences. Dissertationes Math., 321 (1992).Google Scholar
2.Argyros, S.. Banach spaces of the type of Tsirelson. To appear.Google Scholar
3.Beauzamy, B. and Lapreste, J. T.. Modèlesétalés des espaces de Banach, Travaux en Cours (Hermann, Paris, 1984).Google Scholar
4.Bellenot, S., Haydon, R. and Odell, E.. Quasi-reflexive and tree spaces constructed in the spirit of R. C. James. Contemporary Math., 85 (1989), 19–43.CrossRefGoogle Scholar
5.Bessaga, C. and Pelczynski, A.. On bases and unconditional convergence of series in Banach spaces. Studia Math., 17 (1958), 151–164.CrossRefGoogle Scholar
6.Bourgain, J.. On convergent sequences of continuous functions. Bull. Soc. Math. Bel., 32 (1980), 235–249.Google Scholar
7.Giesy, D. P. and James, R. C.. Uniformly non-11 and B-convex Banach spaces. Studia Math., 48 (1973), 61–69.CrossRefGoogle Scholar
8.Haydon, R., Odell, E. and Rosenthal, H.. On certain classes of Baire-1 functions with applications to Banach space theory. In Functional Analysis, Editors Odell, E. and Rosenthal, H., Lectures Notes, 1470 (Springer, 1991), 1–35.Google Scholar
9.James, R. C.. Uniformly non-square Banach spaces. Ann. of Math., 80 (1964), 542–550.CrossRefGoogle Scholar
10.Kechris, A. and Louveau, A.. A classification of Baire class 1 functions. Trans. Amer. Math. Soc., 318 (1990), 209–236.CrossRefGoogle Scholar
11.Lindenstrauss, J. and Tzafriri, L.. Classical Banach spaces, I (Springer, Berlin, 1977).CrossRefGoogle Scholar
12.Schreier, J.. Ein Gegenbeispiel zur Theorie der schwachen Konvergenz. Studia Math., 2 (1930), 58–62.CrossRefGoogle Scholar