On the set of limit points of conditionally convergent series

Szymon Głąb¹ ; Jacek Marchwicki¹

¹ Institute of Mathematics Łódź University of Technology Wólczańska 215 93-005 Łódź, Poland

Studia Mathematica, Tome 237 (2017) no. 3, pp. 221-239

Résumé

Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach space and let $\tau $ be a permutation of the natural numbers. We study the set $\operatorname {LIM}(\sum_{n=1}^\infty x_{\tau (n)})$ of all limit points of the sequence $(\sum_{n=1}^p x_{\tau (n)})_{p=1}^\infty $ of partial sums of the rearranged series $\sum_{n=1}^\infty x_{\tau (n)}$. We give a full characterization of such limit sets in finite-dimensional spaces. Namely, every such limit set in $\mathbb R^m$ is either compact and connected, or closed with all connected components unbounded. On the other hand, each set of one of these types is the limit set of some rearranged conditionally convergent series. Moreover, this characterization does not hold in infinite-dimensional spaces. We show that if $\sum_{n=1}^\infty x_n$ has the Rearrangement Property and $A$ is a closed subset of the closure of the sum range of $\sum_{n=1}^\infty x_n$ and it is $\varepsilon $-chainable for every $\varepsilon \gt 0$, then there is a permutation $\tau $ such that $A=\operatorname {LIM}(\sum_{n=1}^\infty x_{\tau (n)})$.

Détail

DOI : 10.4064/sm8480-10-2016

Keywords: sum infty conditionally convergent series banach space tau permutation natural numbers study set operatorname lim sum infty tau limit points sequence sum tau infty partial sums rearranged series sum infty tau full characterization limit sets finite dimensional spaces namely every limit set mathbb either compact connected closed connected components unbounded other each set these types limit set rearranged conditionally convergent series moreover characterization does infinite dimensional spaces sum infty has rearrangement property closed subset closure sum range sum infty varepsilon chainable every varepsilon there permutation tau operatorname lim sum infty tau