# Rearranging absolutely convergent well-ordered series in Banach spaces

## TL;DR

This paper establishes a general principle for rearranging absolutely convergent well-ordered series in Banach spaces, showing that such rearrangements preserve convergence and sum when indexed by countable ordinals.

## Contribution

It extends classical results on series rearrangements to the setting of Banach spaces with well-ordered series indexed by countable ordinals.

## Key findings

- Rearrangements preserve absolute convergence in Banach spaces.
- Series sum remains unchanged under any countable ordinal reordering.
- Generalizes classical real series results to Banach space context.

## 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.