  Extending the classical Levy-Steinitz rearrangement theorem, which in turn
extended Riemann's theorem, Banaszczyk proved in 1990/93 that a metrizable,
locally convex space is nuclear if and only if the domain of sums of every
convergent series (i.e. the set of all elements in the space which are sums of
a convergent rearrangement of the series) is a translate of a closed subspace
of a special form. In this paper we present an apparently complete analysis of
the domains of convergent series in duals of metrizable spaces or, more
generally, in (DF)-spaces in the sense of Grothendieck.
