On a Theorem of Steinitz and Levy

Gadi Moran
1978 Transactions of the American Mathematical Society  
Let 2"eú)A(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 £ of dimension > 0. Another generalization of Riemann's Theorem [M] can be stated as an instance of the
more » ... n instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval /. This instance generalizes to the Banach space of regulated ¿-valued functions on /, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.
doi:10.2307/1997989 fatcat:rvv7jltij5gbdhw5lt6flyn4fq