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 ${\displaystyle X}$ (that is, ${\displaystyle x_{i}\geq 0}$ for all ${\displaystyle i}$) is called order summable if ${\displaystyle \sup _{n=1,2,\ldots }\sum _{i=1}^{n}x_{i}}$ exists in ${\displaystyle X}$.^[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 ${\displaystyle X}$ is of type ${\displaystyle \ell ^{p}}$ if there exists some ${\displaystyle z\in X}$ and some sequence ${\displaystyle \left(c_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle \ell ^{p}}$ such that ${\displaystyle 0\leq x_{i}\leq c_{i}z}$ for all ${\displaystyle i}$.^[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) – 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

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