$F_{\sigma }$ ideals of perfectly bounded sets

Abstract

Let $\mathbf{x}=(x_n{)}_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\Vert {\sum }_{n\in F}{x}_n\Vert \le M$ for every finite $F\subseteq A$. The collection $B(\mathbf{x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B(\mathbf{x})$ iff there is a non pathological lower semicontinuous submeasure $\phi$ on $\mathbb{N}$ such that $\mathcal{I}=FIN(\phi )=\{A\subseteq \mathbb{N}:\phi (A)<\infty \}$. We address the questions of when $FIN(\phi )$ is a tall ideal and has a Borel selector. We show that in $c_0$ the ideal $B(\mathbf{x})$ is tall iff $(x_n{)}_n$ is weakly null, in which case, it also has a Borel selector.

Substitute by preprint arXiv 2211.01544