On rearrangements of series in locally convex spaces (Q1380998)

This paper deals with the structure of the set of sums of a conditionally convergent series in locally convex metrizable topological vector spaces. The problem goes back to the well-known theorem in analysis due to Riemann asserting that the set of sums \({\mathcal S}_{(a_k)}\) of a conditionally convergent series \(\sum^\infty_{k=1} a_k\), \(a_k\in \mathbb{R}\), under all its convergent rearrangements fills the whole real line \(\mathbb{R}\). This was generalized to the case of finite-dimensional spaces by Levy and Steinitz.