In this article we suppose that E is an ordered Banach space whose positive cone is defined by a countable family $ℱ=f_i|i\in \mathbb{N}$ of positive continuous linear functionals on E, i.e. E₊ = x ∈ E | $f_i\left(x\right)\ge 0$ 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 $x=\left(f_i\left(x\right)\right)$ and we develop a process of successive decompositions of a quasi-interior point of X₊ which at each step gives elements with smaller support....