Order summable

In mathematics, specifically in order theory and functional analysis, a sequence of positive elements $$\left(x_i\right)_{i=1}^{\infty}$$ in a preordered vector space $$X$$ (that is, $$x_i \geq 0$$ for all $$i$$) is called order summable if $$\sup_{n = 1, 2, \ldots} \sum_{i=1}^n x_i$$ exists in $$X$$. For any $$1 \leq p \leq \infty$$, we say that a sequence $$\left(x_i\right)_{i=1}^{\infty}$$ of positive elements of $$X$$ is of type $$\ell^p$$ if there exists some $$z \in X$$ and some sequence $$\left(c_i\right)_{i=1}^{\infty}$$ in $$\ell^p$$ such that $$0 \leq x_i \leq c_i z$$ for all $$i$$.

The notion of order summable sequences is related to the completeness of the order topology.