Study of the Range of Sums of a Vector Series by Multiplying the Rearrangements of a Series by Real Numbers

In this article we consider a new notion connected with the problem of the range of sums of a series in an infinite-dimensional space, multiplication of a rearrangement of a series by a real number. We distinguish some set of rearrangements that admit multiplication by integers. If the sum of a series disturbs after applying one of these rearrangements then the range of sums is unbounded and we can indicate some elements in it. For some subset of rearrangements in this set we prove the impossibility of multiplying a rearrangement by nonintegral numbers.