On the set of limit points of conditionally convergent series

DOI10.4064/SM8480-10-2016zbMATH Open1378.40001arXiv1604.06255OpenAlexW2963648433MaRDI QIDQ2985969

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Publication date: 10 May 2017

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Abstract: Let $su{m}_{n=1}^{i}nfty{x}_n$ be a conditionally convergent series in a Banach space and let $au$ be a permutation of natural numbers. We study the set $operatornameLIM(su{m}_{n=1}^{i}nfty{x}_{au(n)})$ of all limit points of a sequence $(su{m}_{n=1}^p{x}_{au(n)}{)}_{p=1}^{i}nfty$ of partial sums of a rearranged series $su{m}_{n=1}^{i}nfty{x}_{au(n)}$. 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 $su{m}_{n=1}^{i}nfty{x}_n$ has the Rearrangement Property and $A$ is a closed subset of the closure of the $su{m}_{n=1}^{i}nfty{x}_n$ sum range and it is $varepsilon$-chainable for every $varepsilon>0$, then there is a permutation $au$ such that $A=operatornameLIM(su{m}_{n=1}^{i}nfty{x}_{au(n)})$. As a byproduct of this observation we obtain that series having the Rearrangement Property have closed sum ranges.

Full work available at URL: https://arxiv.org/abs/1604.06255

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