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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On a theorem of Steinitz and Levy

Author: Gadi Moran
Journal: Trans. Amer. Math. Soc. 246 (1978), 483-491
MSC: Primary 40A99; Secondary 46B15
MathSciNet review: 515554
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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.

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Keywords: Conditionally convergent series, rearrangement of terms, finite dimensional Banach space
Article copyright: © Copyright 1978 American Mathematical Society