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 $X$.Template:Sfn 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 $i$.Template:Sfn

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

See also

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Bibliography

• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces