Double series (${\sum }_m{\sum }_n{a}_{mn}$)

A double series is a series whose terms carry two indices. For nonnegative terms, define its sum as the supremum of all finite subsums; iterated sums in either order give this same value, possibly infinity. For absolutely summable terms, both orders also agree and give a finite sum. Without these hypotheses, an order of summation must be specified and changing it can change the result. A triangular index set, such as $0\le n\le m$, can be represented by setting the other terms to zero.