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## Studia Mathematica

1999 | 135 | 1 | 83-102
Tytuł artykułu

### On spreading $c_0$-sequences in Banach spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We introduce and study the spreading-(s) and the spreading-(u) property of a Banach space and their relations. A space has the spreading-(s) property if every normalized weakly null sequence has a subsequence with a spreading model equivalent to the usual basis of $c_0$; while it has the spreading-(u) property if every weak Cauchy and non-weakly convergent sequence has a convex block subsequence with a spreading model equivalent to the summing basis of $c_0$. The main results proved are the following: (a) A Banach space X has the spreading-(s) property if and only if for every subspace Y of X and for every pair of sequences (x_n) in Y and $(x*_n)$ in Y*, with(x_n) weakly null in Y and $(x_n*)$ uniformly weakly null in Y* (in the sense of Mercourakis), we have $x*_n(x_n) → 0$ (i.e. X has a hereditary weak Dunford-Pettis property). (b) A Banach space X has the spreading-(u) property if and only if $B_1(X) ⊆ B_{1/4}(X)$ in the sense of the classification of Baire-1 elements of X** according to Haydon-Odell-Rosenthal. (c) The spreading-(s) property implies the spreading-(u) property. Result (c), proved via infinite combinations, connects an internal condition on a Banach space with an external one.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
83-102
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-05-11
poprawiono
1998-10-02
Twórcy
autor
• Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece, vgeorgil@atlas.uoa.gr
Bibliografia
• [B-L] B. Beauzamy et J.-T. Lapresté, Modèles étalés des espaces de Banach, Hermann, Paris, 1984.
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• [B-P] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164.
• [B-S] A. Brunel and L. Sucheston, On B-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299.
• [C] P. Cembranos, The hereditary Dunford-Pettis property on C(K,E), Illinois J. Math. 31 (1987), 365-373.
• [E] J. Elton, Weakly null normalized sequences in Banach spaces, Doctoral Thesis, Yale University, 1978.
• [E-M] P. Erdős and M. Magidor, A note on regular methods of summability and the Banach-Saks property, Proc. Amer. Math. Soc. 59 (1976), 232-234.
• [F1] V. Farmaki, On Baire-1/4 functions and spreading models, Mathematika 41 (1994), 251-265.
• [F1] V. Farmaki, Classifications of Baire-1 functions and $c_0$-spreading models, Trans. Amer. Math. Soc. 345 (1994), 819-831.
• [F1] V. Farmaki, On Baire-1/4 functions, ibid. 348 (1996), 4023-4041.
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• [G] S. Guerre-Delabrière, Classical Sequences in Banach Spaces, Monographs and Textbooks Pure Appl. Math. 166, Dekker, 1992.
• [H-O-R] R. Haydon, E. Odell and H. Rosenthal, On certain classes of Baire-1 functions with applications to Banach space theory, in: Lecture Notes in Math. 1470, Springer, 1991, 1-35.
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• [K-O] H. Knaust and E. Odell, On $c_0$-sequences in Banach spaces, Israel J. Math. 67 (1989), 153-169.
• [M] S. Mercourakis, On Cesàro summable sequences of continuous functions, Mathematika 42 (1995), 87-104.
• [O-1] E. Odell, Applications of Ramsey theorems to Banach space theory, in: Notes in Banach Spaces, H. E. Lacey (ed.), Univ. of Texas Press, 1980, 379-404.
• [O-1] E. Odell, On Schreier unconditional sequences, in: Contemp. Math. 144, Amer. Math. Soc., 1993, 197-201.
• [O-R] E. Odell and H. Rosenthal, A double dual characterization of separable Banach spaces containing $ℓ^1$, Israel J. Math. 20 (1975), 375-384.
• [P] A. Pełczyński, A connection between weakly unconditional convergence and weakly completeness of Banach spaces, Bull. Acad. Polon. Sci. 6 (1958), 251-253.
• [R1] H. Rosenthal, Weakly independent sequences and the Banach-Saks property, Bull. London Math. Soc. 8 (1976), 22-24.
• [R1] H. Rosenthal, A characterization of Banach spaces containing $c_0$, J. Amer. Math. Soc. 7 (1994), 707-748.
• [R1] H. Rosenthal, Differences of bounded semi-continuous functions, I, to appear.
• [S] I. Singer, Bases in Banach Spaces I, Springer, 1970.
• [T] B. S. Tsirelson, Not every Banach space contains $ℓ_p$ or $c_0$, Functional Anal. Appl. 8 (1974), 138-141.
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Bibliografia
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