An approach to Schreier's space
Jesús M. F. Castillo, Manuel González · Extracta Mathematicae · 1991
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 I 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.