On a theorem of Steinitz and Levy

Abstract:

Let $\sum \nolimits _{n \in \omega } {h(n)}$ be a conditionally convergent series in a real Banach space B. Let $S(h)$ denote the set of sums of the convergent rearrangements of this series. A well-known theorem of Riemann states that $S(h) = B$ if $B = R$, the reals. A generalization of Riemann’s Theorem, due independently to Levy [L] and Steinitz [S], states that if B is finite dimensional, then $S(h)$ is a linear manifold in B of dimension $> 0$. Another generalization of Riemann’s Theorem [M] can be stated as an instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated B-valued functions on I, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.