Order summable explained

In mathematics, specifically in order theory and functional analysis, a sequence of positive elements

\left(x _{i\right)} ^{inf ty} ^{i =1} in a preordered vector space X (that is, x _{i} \geq 0 for all i ) is called order summable if \sup _{n} \sum ^{n} _{i =1} x _{i} exists in X . For any 1 \leq p \leq inf ty , we say that a sequence \left(x _{i\right)} ^{inf ty} ^{i =1} of positive elements of X is of type \ell ^{p} if there exists some z \in X and some sequence \left(c _{i\right)} ^{inf ty} ^{i =1} 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.