On a theorem of Steinitz and Levy

Let ∑ n ∈ ω h ( n ) \sum \nolimits _{n\,\, \in \,\omega } {h(n)} be a conditionally convergent series in a real Banach space B. Let S ( h ) 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 S(h)\, = \,B if B = R 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 ) S(h) is a linear manifold in B of dimension > 0 > \,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.