Order summable

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

The notion of order summable sequences is related to the completeness of the order topology.

References

1 2 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.

[FOOTNOTESchaeferWolff1999230–234-1] 1 2 Schaefer & Wolff 1999, pp. 230–234.

[1]

v t e Ordered topological vector spaces
Basic concepts	Ordered vector space Partially ordered space Riesz space Order topology Order unit Positive linear operator Topological vector lattice Vector lattice
Types of orders/spaces	AL-space AM-space Archimedean Banach lattice Fréchet lattice Locally convex vector lattice Normed lattice Order bound dual Order dual Order complete Regularly ordered
Types of elements/subsets	Band Cone-saturated Lattice disjoint Dual/Polar cone Normal cone Order complete Order summable Order unit Quasi-interior point Solid set Weak order unit
Topologies/Convergence	Order convergence Order topology
Operators	Positive State
Main results	Freudenthal spectral

See also

References

Bibliography