Study of domain of sums of vector-valued series in terms of multiplication of a permutation of the series by real numbers (Q2773633)

The article is mainly devoted to the following problem: Let \(\sum_{k=1}^{\infty} x_k\) be a (conditionally) convergent series in a normed space \(X\). What can be said about the set of sums of the convergent rearrangements of \(\sum_{k=1}^{\infty} x_k\)? NEWLINENEWLINENEWLINEThe author introduces a new notion of a multiplication of a rearrangement of a series by a real number and applies this notion to the above problem. In particular, she shows that if the sum of a series may be changed under a permutation then NEWLINENEWLINENEWLINE(i) the domain of sums is unbounded and NEWLINENEWLINENEWLINE(ii) some elements of the domain of sums may be given explicitly.