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      Rearranging absolutely convergent well-ordered series in Banach spaces

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          Abstract

          Reordering the terms of a series is a useful mathematical device, and much is known about when it can be done without affecting the convergence or the sum of the series. For example, if a series of real numbers absolutely converges, we can add the even-indexed and odd-indexed terms separately, or arrange the terms in an infinite two-dimensional table and first compute the sum of each column. The possibility of even more intricate re-orderings prompts us to find a general underlying principle. We identify such a principle in the setting of Banach spaces, where we consider well-ordered series with indices beyond {\omega}, but strictly under {\omega}_1 . We prove that for every absolutely convergent well-ordered series indexed by a countable ordinal, if the series is rearranged according to any countable ordinal, then the absolute convergence and the sum of the series remain unchanged.

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          Most cited references 4

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          Absolute and Unconditional Convergence in Normed Linear Spaces

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            Well-Ordered Sub-Series of General Series

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              On unconditional convergence in normed vector spaces

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                Author and article information

                Journal
                23 February 2019
                Article
                1902.08846

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                03E10, 40A05
                math.LO

                Logic & Foundation

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