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1999 | 135 | 1 | 83-102
Tytuł artykułu

On spreading $c_0$-sequences in Banach spaces

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
Czasopismo
Rocznik
Tom
135
Numer
1
Strony
83-102
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-05-11
poprawiono
1998-10-02
Twórcy
  • Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece, vgeorgil@atlas.uoa.gr
Bibliografia
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  • [F1] V. Farmaki, On Baire-1/4 functions, ibid. 348 (1996), 4023-4041.
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  • [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.
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Bibliografia
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