Rearranging series. The Lévy-Steinitz theorem (Q2878049)

Let \(\sum x_n\) be a series in a topological vector space \(X\). Its sum-set \(S(\sum x_n)\) is the set of those \(x \in X\) for which for some rearrangement \(\pi: \mathbb N \to \mathbb N\) the corresponding rearranged series \(\sum_{n = 1}^\infty x_{\pi(n)}\) converges to \(x\). The classical Lévy-Steinitz theorem says that, for a conditionally convergent series \(\sum x_n\) in a finite-dimensional space, the corresponding set \(S(\sum x_n)\) is a shifted subspace.NEWLINENEWLINEThe paper under review is an expanded version of a talk given by the author on the First Joint Congress of the Spanish Royal Mathematical Society and the Mexican Mathematical Society. The paper contains the history of the question, a number of nice examples, a sketch of the proof of the Lévy-Steinitz theorem and a survey of infinite-dimensional generalizations, both for Banach spaces and for topological vector spaces. Some open problems are mentioned.