Order Summable

In mathematics, specifically in order theory and functional analysis, a sequence of positive elements $\left(x_i\right)_^$ in a preordered vector space $X$ (that is, $x_i \geq 0$ for all $i$) is called order summable if $\sup_ \sum_^n x_i$ exists in $X$.
For any $1 \leq p \leq \infty$, we say that a sequence $\left(x_i\right)_^$ of positive elements of $X$ is of type $\ell^p$ if there exists some $z \in X$ and some sequence $\left(c_i\right)_^$ 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

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References

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{{Ordered topological vector spaces

Functional analysis