46b15, 54a20, 54e15

In functional analysis the notion of a summable family (with sum x) is well-known. If (x i) i2I is a family of points from a normed space, (x i) is called summable to x ii for each " > 0 there exits a nite set F 0 I such that kx ? P i2F x i k < " for each nite set F, F 0 F I. But we can interprete this deenition as the convergence of a suitable net. In a normed commutative group we characterize the fact that this net is a Cauchy net.