# An approach to Schreier's space.

Extracta Mathematicae (1991)

• Volume: 6, Issue: 2-3, page 166-169
• ISSN: 0213-8743

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## Abstract

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In 1930, J. Schreier [10] introduced the notion of admissibility in order to show that the now called weak-Banach-Saks property does not hold in every Banach space. A variation of this idea produced the Schreier's space (see [1],[2]). This is the space obtained by completion of the space of finite sequences with respect to the following norm:||x||S = sup(A admissible) ∑j ∈ A |xj|,where a finite sub-set of natural numbers A = {n1 &lt; ... &lt; nk} is said to be admissible if k ≤ n1.In this extract we collect the basic properties of S, which can be considered mainly folklore, and show how this space can be used to provide counter examples to the three-space problem for several properties such as: Dunford-Pettis and Hereditary Dunford-Pettis, weak p-Banach-Saks, and Sp.

## How to cite

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Fernández Castillo, Jesús M., and González, Manuel. "An approach to Schreier's space.." Extracta Mathematicae 6.2-3 (1991): 166-169. <http://eudml.org/doc/39945>.

@article{FernándezCastillo1991,
abstract = {In 1930, J. Schreier [10] introduced the notion of admissibility in order to show that the now called weak-Banach-Saks property does not hold in every Banach space. A variation of this idea produced the Schreier's space (see [1],[2]). This is the space obtained by completion of the space of finite sequences with respect to the following norm:||x||S = sup(A admissible) ∑j ∈ A |xj|,where a finite sub-set of natural numbers A = \{n1 &lt; ... &lt; nk\} is said to be admissible if k ≤ n1.In this extract we collect the basic properties of S, which can be considered mainly folklore, and show how this space can be used to provide counter examples to the three-space problem for several properties such as: Dunford-Pettis and Hereditary Dunford-Pettis, weak p-Banach-Saks, and Sp.},
author = {Fernández Castillo, Jesús M., González, Manuel},
journal = {Extracta Mathematicae},
keywords = {Espacios de Banach; Espacios normados; Propiedad de Dunford-Pettis; Problema de tres espacios},
language = {eng},
number = {2-3},
pages = {166-169},
title = {An approach to Schreier's space.},
url = {http://eudml.org/doc/39945},
volume = {6},
year = {1991},
}

TY - JOUR
AU - Fernández Castillo, Jesús M.
AU - González, Manuel
TI - An approach to Schreier's space.
JO - Extracta Mathematicae
PY - 1991
VL - 6
IS - 2-3
SP - 166
EP - 169
AB - In 1930, J. Schreier [10] introduced the notion of admissibility in order to show that the now called weak-Banach-Saks property does not hold in every Banach space. A variation of this idea produced the Schreier's space (see [1],[2]). This is the space obtained by completion of the space of finite sequences with respect to the following norm:||x||S = sup(A admissible) ∑j ∈ A |xj|,where a finite sub-set of natural numbers A = {n1 &lt; ... &lt; nk} is said to be admissible if k ≤ n1.In this extract we collect the basic properties of S, which can be considered mainly folklore, and show how this space can be used to provide counter examples to the three-space problem for several properties such as: Dunford-Pettis and Hereditary Dunford-Pettis, weak p-Banach-Saks, and Sp.
LA - eng
KW - Espacios de Banach; Espacios normados; Propiedad de Dunford-Pettis; Problema de tres espacios
UR - http://eudml.org/doc/39945
ER -

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