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 < ... < 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.
@article{urn:eudml:doc:39945,
title = {An approach to Schreier's space.},
journal = {Extracta Mathematicae},
volume = {6},
year = {1991},
pages = {166-169},
mrnumber = {MR1185369},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39945}
}
Fernández Castillo, Jesús M.; González, Manuel. An approach to Schreier's space.. Extracta Mathematicae, Tome 6 (1991) pp. 166-169. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39945/