Order summable

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

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.