Order summable

In mathematics, specifically in order theory and functional analysis, a sequence of positive elements ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ in a preordered vector space X (i.e. x_i ≥ 0 for all i) is called order summable if ${\displaystyle \sup \left\{\sum _{i=1}^{n}x_{i}:n=1,2,\ldots \right\}}$ exists in X.[1] For any ${\displaystyle 1\leq p\leq \infty }$, we say that a sequence ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ of positive elements of X is of type ${\displaystyle l^{p}}$ if there exists some z in X and some sequence ${\displaystyle \left(c_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle l^{p}}$ such that ${\displaystyle 0\leq x_{i}\leq c_{i}z}$ for all i.[1]

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