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In mathematics, specifically in order theory and functional analysis, a sequence of positive elements i = 1 ∞ {\displaystyle \left_{i=1}^{\infty }} in a preordered vector space X {\displaystyle X} is called order summable if sup n = 1 , 2 , … ∑ i = 1 n x i {\displaystyle \sup _{n=1,2,\ldots }\sum _{i=1}^{n}x_{i}} exists in X {\displaystyle X}. For any 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } , we say that a sequence i = 1 ∞ {\displaystyle \left_{i=1}^{\infty }} of positive elements of X {\displaystyle X} is of type ℓ p {\displaystyle \ell ^{p}} if there exists some z ∈ X {\displaystyle z\in X} and some sequence i = 1 ∞ {\displaystyle \left_{i=1}^{\infty }} in ℓ p {\displaystyle \ell ^{p}} such that 0 ≤ x i ≤ c i z {\displaystyle 0\leq x_{i}\leq c_{i}z} for all i {\displaystyle i}.
The notion of order summable sequences is related to the completeness of the order topology.