On a theorem of Steinitz and Levy
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 by Gadi Moran PDF
 Trans. Amer. Math. Soc. 246 (1978), 483491 Request permission
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 wellknown 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 LevySteinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated Bvalued functions on I, where B is finite dimensional, implying a generalization of the LevySteinitz Theorem.References

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Additional Information
 © Copyright 1978 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 246 (1978), 483491
 MSC: Primary 40A99; Secondary 46B15
 DOI: https://doi.org/10.1090/S00029947197805155547
 MathSciNet review: 515554