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In mathematics, specifically in [[order theory]] and [[functional analysis]], a sequence of positive elements <math>\left(x_i\right)_{i=1}^{\infty}</math> in a [[Ordered vector space|preordered vector space]] <math>X</math> (that is, <math>x_i \geq 0</math> for all <math>i</math>) is called '''order summable''' if <math>\sup_{n = 1, 2, \ldots} \sum_{i=1}^n x_i</math> exists in <math>X</math>.{{sfn|Schaefer|Wolff|1999|pp=230–234}} 
For any <math>1 \leq p \leq \infty</math>, we say that a sequence <math>\left(x_i\right)_{i=1}^{\infty}</math> of positive elements of <math>X</math> is of '''type <math>\ell^p</math>''' if there exists some <math>z \in X</math> and some sequence <math>\left(c_i\right)_{i=1}^{\infty}</math> in <math>\ell^p</math> such that <math>0 \leq x_i \leq c_i z</math> for all <math>i</math>.{{sfn|Schaefer|Wolff|1999|pp=230–234}} 

The notion of order summable sequences is related to the completeness of the [[Order topology (functional analysis)|order topology]]. 

== See also ==

* {{annotated link|Ordered topological vector space}}
* {{annotated link|Order topology (functional analysis)}}
* {{annotated link|Ordered vector space}}
* {{annotated link|Vector lattice}}

== References ==
{{reflist}}

==Bibliography==

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->

{{Functional analysis}}
{{Ordered topological vector spaces}}

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[[Category:Functional analysis]]