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