In mathematics, specifically in order theory and functional analysis, a sequence of positive elements ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in a preordered vector space X {\displaystyle X} (that is, x i ≥ 0 {\displaystyle x_{i}\geq 0} for all i {\displaystyle i} ) 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 ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{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 ( c i ) i = 1 ∞ {\displaystyle \left(c_{i}\right)_{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.
See also Ordered topological vector space 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
References
Bibliography Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
