Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 40A99, 46B15

Retrieve articles in all journals with MSC: 40A99, 46B15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515554-7
PII: S 0002-9947(1978)0515554-7
Keywords: Conditionally convergent series, rearrangement of terms, finite dimensional Banach space
Article copyright: © Copyright 1978 American Mathematical Society