Order summable
In mathematics, specifically in order theory and functional analysis, a sequence of positive elements [math]\displaystyle{ \left(x_i\right)_{i=1}^{\infty} }[/math] in a preordered vector space [math]\displaystyle{ X }[/math] (that is, [math]\displaystyle{ x_i \geq 0 }[/math] for all [math]\displaystyle{ i }[/math]) is called order summable if [math]\displaystyle{ \sup_{n = 1, 2, \ldots} \sum_{i=1}^n x_i }[/math] exists in [math]\displaystyle{ X }[/math].[1] For any [math]\displaystyle{ 1 \leq p \leq \infty }[/math], we say that a sequence [math]\displaystyle{ \left(x_i\right)_{i=1}^{\infty} }[/math] of positive elements of [math]\displaystyle{ X }[/math] is of type [math]\displaystyle{ \ell^p }[/math] if there exists some [math]\displaystyle{ z \in X }[/math] and some sequence [math]\displaystyle{ \left(c_i\right)_{i=1}^{\infty} }[/math] in [math]\displaystyle{ \ell^p }[/math] such that [math]\displaystyle{ 0 \leq x_i \leq c_i z }[/math] for all [math]\displaystyle{ i }[/math].[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
References
- ↑ 1.0 1.1 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.
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