Summation of families over arbitrary index sets

Summary

Let $(x_i)_{i \in I}$ be a family indexed by an arbitrary set $I$. It is summable when its finite partial sums approach a limit as the finite set of indices grows. This notion of sum uses no order on $I$, so it works for index sets such as $\N \times \N$, $\Z$ or uncountable sets. For real or complex terms indexed by $\N$, summability is the same as absolute convergence. The exact definition is in summable family.