# Ideal boundedness of subseries and rearrangements in Banach spaces vs Banach spaces possessing a copy of $c_0$

## TL;DR

This paper characterizes Banach spaces that do not contain a copy of c_0 by the meagerness of sets coding bounded subseries and rearrangements, using Bessaga-Pe{
}lczy{\'n}ski's theorem and Baire ideal concepts.

## Contribution

It establishes a new equivalence between the absence of c_0 in Banach spaces and the meagerness of certain sets related to series rearrangements.

## Key findings

- Spaces without c_0 have meager sets for bounded subseries and rearrangements.
- Uses Bessaga-Pe{
}lczy{\'n}ski c_0-Theorem and Baire ideal theory.
- Provides a characterization involving ideal boundedness and Talagrand's theorem.

## Abstract

Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are meager. We use Bessaga-Pe{\l}czy{\'n}ski $c_0-$Theorem and concept of uniformly unconditionally bounded series. Moreover we prove similar result for the ideal boundedness for a class of Baire ideals using Talagrand's characterisation.