Let be a normalized vector space. Be an index set and a family. Be .
The family is called summable to if and only if
applies. So if for each a finite subset can be found in such a way that for all finite supersets that lie in, the sum in the norm deviates from less than .
As with series, absolute summability can also be defined. The family is called absolutely summable to and only if is summable to one .
Ultimately, a family is called Cauchy-summable if and only if
applies.
Remarks
Absolute summability implies summability. The reverse is generally not true.
If a family can be totaled, each subfamily can also be totaled. Summability is therefore a stronger criterion than simple convergence of series.
Cauchy summability follows from summability. The reverse is true in Banach spaces . Cauchy summability is often easier to test.
Be summable to and summable to and a scalar. Then applies .