Order summable

In mathematics, specifically in order theory and functional analysis, a sequence of positive elements $\left(x_{i}\right)_{i=1}^{\infty }$ in a preordered vector space $X$ (that is, $x_{i}\geq 0$ for all $i$) is called order summable if $\sup _{n=1,2,\ldots }\sum _{i=1}^{n}x_{i}$ exists in $X$. For any $1\leq p\leq \infty$, we say that a sequence $\left(x_{i}\right)_{i=1}^{\infty }$ of positive elements of $X$ is of type $\ell ^{p}$ if there exists some $z\in X$ and some sequence $\left(c_{i}\right)_{i=1}^{\infty }$ in $\ell ^{p}$ such that $0\leq x_{i}\leq c_{i}z$ for all $i$.

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

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.