Positive bases in ordered subspaces with the Riesz decomposition property

Vasilios Katsikis ¹   ; Ioannis A. Polyrakis ¹

¹ Department of Mathematics National Technical University of Athens Zografou Campus 157 80, Athens, Greece

Studia Mathematica, Tome 174 (2006) no. 3, pp. 233-253

Résumé

In this article we suppose that $E$ is an ordered Banach space whose positive cone is defined by a countable family $\mathcal F = \{f_i\mid i\in \mathbb{N}\}$ of positive continuous linear functionals on $E$, i.e. $E_+ = \{x\in E\mid f_i(x)\geq 0 \hbox{ for each }i\}$, and we study the existence of positive (Schauder) bases in ordered subspaces $X$ of $E$ with the Riesz decomposition property. We consider the elements $x$ of $E$ as sequences $x=(f_i(x))$ and we develop a process of successive decompositions of a quasi-interior point of $X_+$ which at each step gives elements with smaller support. As a result we obtain elements of $X_+$ with minimal support and we prove that they define a positive basis of $X$ which is also unconditional. In the first section we study ordered normed spaces with the Riesz decomposition property.

Détail

DOI : 10.4064/sm174-3-2

Keywords: article suppose ordered banach space whose positive cone defined countable family mathcal mid mathbb positive continuous linear functionals nbsp mid geq hbox each study existence positive schauder bases ordered subspaces riesz decomposition property consider elements sequences develop process successive decompositions quasi interior point which each step gives elements smaller support result obtain elements minimal support prove define positive basis which unconditional first section study ordered normed spaces riesz decomposition property