On spreading c 0 -sequences in Banach spaces

Vassiliki Farmaki

Studia Mathematica (1999)

  • Volume: 135, Issue: 1, page 83-102
  • ISSN: 0039-3223

Abstract

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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 (xn) in Y and ( x * n ) in Y*, with(xn) 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.

How to cite

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Farmaki, Vassiliki. "On spreading $c_0$-sequences in Banach spaces." Studia Mathematica 135.1 (1999): 83-102. <http://eudml.org/doc/216645>.

@article{Farmaki1999,
abstract = {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 (xn) in Y and $(x*_n)$ in Y*, with(xn) 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.},
author = {Farmaki, Vassiliki},
journal = {Studia Mathematica},
keywords = {spreading-; spreading-},
language = {eng},
number = {1},
pages = {83-102},
title = {On spreading $c_0$-sequences in Banach spaces},
url = {http://eudml.org/doc/216645},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Farmaki, Vassiliki
TI - On spreading $c_0$-sequences in Banach spaces
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 1
SP - 83
EP - 102
AB - 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 (xn) in Y and $(x*_n)$ in Y*, with(xn) 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.
LA - eng
KW - spreading-; spreading-
UR - http://eudml.org/doc/216645
ER -

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