# Article

 Title: Characterizations of spreading models of $l^1$ (English) Author: Kiriakouli, P. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 41 Issue: 1 Year: 2000 Pages: 79-95 . Category: math . Summary: Rosenthal in [11] proved that if $(f_{k})$ is a uniformly bounded sequence of real-valued functions which has no pointwise converging subsequence then $(f_{k})$ has a subsequence which is equivalent to the unit basis of $l^{1}$ in the supremum norm. Kechris and Louveau in [6] classified the pointwise convergent sequences of continuous real-valued functions, which are defined on a compact metric space, by the aid of a countable ordinal index $\gamma$''. In this paper we prove some local analogues of the above Rosenthal 's theorem (spreading models of $l^{1}$) for a uniformly bounded and pointwise convergent sequence $(f_{k})$ of continuous real-valued functions on a compact metric space for which there exists a countable ordinal $\xi$ such that $\gamma ((f_{n_{k}}))> \omega^{\xi}$ for every strictly increasing sequence $(n_{k})$ of natural numbers. Also we obtain a characterization of some subclasses of Baire-1 functions by the aid of spreading models of $l^{1}$. (English) Keyword: uniformly bounded sequences of continuous real-valued functions Keyword: convergence index Keyword: spreading models of $l^{1}$ Keyword: Baire-1 functions MSC: 46B20 MSC: 46B99 MSC: 46E15 MSC: 46E99 MSC: 54C35 idZBL: Zbl 1039.46010 idMR: MR1756928 . Date available: 2009-01-08T18:58:40Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119142 . Reference: [1] Alspach D., Argyros S.: Complexity of weakly null sequences.Dissertations Mathematicae CCCXXI (1992), 1-44. Zbl 0787.46009, MR 1191024 Reference: [2] Alspach D., Odell E.: Averaging null sequences.Lecture Notes in Math. 1332, Springer, Berlin, 1988. MR 0967092 Reference: [3] Argyros S.A., Mercourakis S., Tsarpalias A.: Convex unconditionality and summability of weakly null sequences.Israel J. Math. 107 (1998), 157-193. Zbl 0942.46007, MR 1658551 Reference: [4] Bourgain J.: On convergent sequences of continuous functions.Bull. Soc. Math. Belg. Ser. B 32 (1980), 235-249. Zbl 0474.54008, MR 0682645 Reference: [5] Haydon R., Odell E., Rosenthal H.: On certain classes of Baire-1 functions with applications to Banach space theory.Longhorn Notes, The University of Texas at Austin, Funct. Anal. Sem. 1987-89. Zbl 0762.46006 Reference: [6] Kechris A.S., Louveau A.: A classification of Baire class 1 functions.Trans. Amer. Math. Soc. 318 (1990), 209-236. Zbl 0692.03031, MR 0946424 Reference: [7] Kiriakouli P.: Namioka spaces, Baire-1 functions, Combinatorial principles of the type of Ramsey and their applications in Banach spaces theory (in Greek).Doctoral Dissertation, Athens Univ., 1994. Reference: [8] Kiriakouli P.: Classifications and characterizations of Baire-1 functions.Comment. Math. Univ. Carolinae 39.4 (1998), 733-748. MR 1715462 Reference: [9] Kiriakouli P.: On combinatorial theorems with applications to Banach spaces theory.preprint, 1994. Reference: [10] Mercourakis S., Negrepontis S.: Banach spaces and topology II.Recent Progress in General Topology, M. Hušek and J. van Mill, eds., Elsevier Science Publishers B.V., 1992, pp.495-536. Zbl 0832.46005 Reference: [11] Rosenthal H.P.: A characterization of Banach spaces containing $l^{1}$.Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413. MR 0358307 .

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