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. xi ≥ 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.
See also
Ordered topological vector space
Order topology (functional analysis)
Ordered vector space
Vector lattice
References
Schaefer 1999, pp. 230–234.
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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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