Order summable

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. x_i ≥ 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

1. Schaefer & Wolff 1999, pp. 230–234.

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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.