  We investigate ideals of the form $\{A \subseteq \omega\colon \sum_{n\in A}
x_n$ is unconditionally convergent $\}$, where $(x_n)_{n\in\omega}$ is a
sequence in a Polish group or in a Banach space. If an ideal on $\omega$ can be
seen in this form for some sequence in $X$, then we say that it is
representable in $X$.
  After numerous examples we show the following theorems: (1) An ideal is
representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An
ideal is representable in a Banach space iff it is a non-pathological analytic
P-ideal.
  We focus on the family of ideals representable in $c_0$. We prove that the
trace of the null ideal, Farah's ideal, and Tsirelson ideals are not
representable in $c_0$, and that a tall $F_\sigma$ P-ideal is representable in
$c_0$ iff it is a summable ideal. Also, we provide an example of a peculiar
ideal which is representable in $\ell_1$ but not in $\mathbb{R}$.
  Finally, we summarize some open problems of this topic.
