Harry Poppe
A theorem on summable families in normed groups.
The paper is published: Rostocker Mathematische Kolloquium, Rostock. Math. Kolloq. 49, 51-56 (1995)

MSC:

46B15 Summability and bases, See also {46A35}

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)

Abstract: In functional analysis the notion of a summable family (with sum x) is
wellknown. If (x_i)_{i\in I} is a family of points from a normed space,
(x_i) is called summable to x iff for each \epsilon>0 there exists a
finite set F_0\subset I such that ||x-\Sum_{i\in F}x_i||<\epsilon for
each finite set F, F_0\subset F\subset I. But we can interprete this
definition as the convergence of a suitable net. In a normed commutative
group we charaterize the fact that this net is a Cauchy net.
Keywords: commutative topological group, summable family, special nets, normed (commutative) group, Cauchy nets