  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.
