On the set of limit points of conditionally convergent series

2016/04/21 by Głab, Szymon, Marchwicki, Jacek
#40A05 #46B15 #46B20 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1604.06255

Abstract

Let ∑_n=1^∞ x_n be a conditionally convergent series in a Banach space and let τ be a permutation of natural numbers. We study the set LIM(∑_n=1^∞ x_τ(n)) of all limit points of a sequence (∑_n=1^p x_τ(n))_p=1^∞ of partial sums of a rearranged series ∑_n=1^∞ x_τ(n). We give full characterization of limit sets in finite dimensional spaces. Namely, a limit set in 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 ∑_n=1^∞ x_n has the Rearrangement Property and A is a closed subset of the closure of the ∑_n=1^∞ x_n sum range and it is ε-chainable for every ε>0, then there is a permutation τ such that A=LIM(∑_n=1^∞ x_τ(n)). As a byproduct of this observation we obtain that series having the Rearrangement Property have closed sum ranges.

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