| 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 |
| . |
