Order summable

In mathematics, specifically in order theory and functional analysis, a sequence of positive elements ${\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ in a preordered vector space $X$ (that is, $x_i\ge 0$ for all $i$) is called order summable if $\underset{n=1,2,\dots }{\sup }\sum _{i=1}^n{x}_i$ exists in $X$. For any $1\le p\le \mathrm{\infty }$, we say that a sequence ${\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ of positive elements of $X$ is of type ${\ell }^p$ if there exists some $z\in X$ and some sequence ${\left(c_i\right)}_{i=1}^{\mathrm{\infty }}$ in ${\ell }^p$ such that $0\le x_i\le c_{i}z$ for all $i$.

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

See also

• Order topology (functional analysis) – Topology of an ordered vector space
• Ordered vector space – Vector space with a partial order
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets