On the set of limit points of conditionally convergent series

Abstract

Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach space and let $\tau$ be a permutation of natural numbers. We study the set $\operatorname{LIM}(\sum_{n=1}^\infty x_{\tau(n)})$ of all limit points of a sequence $(\sum_{n=1}^p x_{\tau(n)})_{p=1}^\infty$ of partial sums of a rearranged series $\sum_{n=1}^\infty x_{\tau(n)}$. We give full characterization of limit sets in finite dimensional spaces. Namely, a limit set in $\mathbb{R}^m$ is either compact and connected or it is closed and all its connected components are unbounded. On the other hand each set of one of these types is a 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 $\sum_{n=1}^\infty x_n$ sum range and it is $\varepsilon$-chainable for every $\varepsilon>0$, then there is a permutation $\tau$ such that $A=\operatorname{LIM}(\sum_{n=1}^\infty x_{\tau(n)})$. As a byproduct of this observation we obtain that series having the Rearrangement Property have closed sum ranges.