On the set of limit points of conditionally convergent series

TL;DR

This paper characterizes the set of limit points of rearranged conditionally convergent series in finite and infinite dimensional spaces, revealing a full description in finite dimensions and conditions for certain sets to be limit sets.

Contribution

It provides a complete characterization of limit sets in finite-dimensional spaces and explores conditions under which certain sets are limit sets in Banach spaces.

Findings

In finite dimensions, limit sets are either compact and connected or closed with unbounded connected components.

In infinite dimensions, the characterization does not hold.

Series with the Rearrangement Property can have their limit sets precisely described as certain chainable sets.

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.