A note on multiplier convergent series

Abstract

Given a topological ring R and F⊂RN a (formal) series ∑n∈Nxn in a topological R-module E is Fmultiplier convergent in E (respectively Fmultiplier Cauchy in E) provided that the sequence {∑i=0nr(i)xi:n∈N} of partial sums converges (respectively, is a Cauchy sequence) for every sequence function r∈F. In this paper we investigate for which G⊂RN every F multiplier convergent (Cauchy) series is also G multiplier convergent (Cauchy). We obtain some general theorems about the Cauchy version of this problem. In particular, we prove that every ZN multiplier Cauchy series is already RN multiplier Cauchy in every topological vector space. On the other hand, we construct examples that in particular show that a ZN multiplier convergent series need not to be even QN multiplier convergent and that there are topological vector spaces containing non-trivial QN multiplier convergent series that do not contain non-trivial RN convergent series. As a consequence of this example, there are topological vector spaces containing the topological group QN (and thus ZN and Z(N) as well) that do not contain the topological vector space RN. On the contrary, it was proved in [3], that a sequentially complete topological vector space that contains the topological group Z(N) must already contain the topological vector space RN. Hence our example demonstrates, that in the latter result, the condition of sequential completeness can not be weakened by assuming that the space in question contains the topological group ZN (which is the sequential completion of Z(N)).