The Levy-Steinitz rearrangement theorem for duals of metrizable spaces

Extending the Levy-Steinitz rearrangement theorem in Rn, 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 sums of convergent series in duals of metrizable spaces or, more generally, in (DF)-spaces in the sense of Grothendieck. Introduction For a convergent series ∑ (uk) in a locally convex space E the domain of sums S (∑ (uk) ) is the set of all x ∈ E which can be obtained as the sum of a convergent rearrangement of ∑ (uk). In terms of this notion Riemann’s famous rearrangement theorem states that in the real line R the domains of sums are either single points or coincide with the whole line. Later Levy [L] and Steinitz [S] extended Riemann’s result to finite dimensional spaces: domains of sums in Rn are affine subspaces – more precisely, for each convergent series ∑ (uk) in Rn