# On spreading ${c}_{0}$-sequences in Banach spaces

Studia Mathematica (1999)

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

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topFarmaki, 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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