Positive bases in ordered subspaces with the Riesz decomposition property
Volume 174 / 2006
Abstract
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.
