Positive Bases in Ordered Subspaces with the Riesz Decomposition Property

In this article we suppose that E is an ordered Banach space the positive cone of which is defined by a countable familyF={fi|i ∈ N} of positive continuous linear functionals of E, i.e. E+ = {x ∈ E | fi(x) ≥ 0, for each i} and we study the existence of positive (Schauder) bases in the ordered subspaces X of E with the Riesz decomposition property. So we consider the elements x of E as sequences x = (fi(x)) and we develop a process of successive decompositions of a quasi-interior point of X+ which in any step gives elements with smaller support. So we obtain elements of X+ with minimal support and we prove that these elements define a positive basis of X which is also unconditional. In the first section of this article we study ordered normed spaces with the Riesz decomposition property.