A note on multiplier convergent series

Given a topological ring R   and F⊂RNF⊂RN a (formal) series ∑n∈Nxn∑n∈Nxn in a topological R-module E   is FFmultiplier convergent in E   (respectively FFmultiplier 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∈Fr∈F. In this paper we investigate for which G⊂RNG⊂RN every FF multiplier convergent (Cauchy) series is also GG multiplier convergent (Cauchy). We obtain some general theorems about the Cauchy version of this problem. In particular, we prove that every ZNZN multiplier Cauchy series is already RNRN multiplier Cauchy in every topological vector space. On the other hand, we construct examples that in particular show that a ZNZN multiplier convergent series need not to be even QNQN multiplier convergent and that there are topological vector spaces containing non-trivial QNQN multiplier convergent series that do not contain non-trivial RNRN convergent series. As a consequence of this example, there are topological vector spaces containing the topological group QNQN (and thus ZNZN and Z(N)Z(N) as well) that do not contain the topological vector space RNRN. On the contrary, it was proved in [3], that a sequentially complete topological vector space that contains the topological group Z(N)Z(N) must already contain the topological vector space RNRN. 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 ZNZN (which is the sequential completion of Z(N)Z(N)).