On Rearrangements of Series in Locally Convex Spaces

摘要

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 S_((a_k)) of a conditionally convergent series ∑ from k=1 to ∞ of a_k, a_k ∈ R, under all its convergent rearrangements fills the whole real line R. This was generalized to the case of finite-dimensional spaces by Levy and Steinitz.