Lévy–Steinitz theorem
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In mathematics, the Lévy–Steinitz theorem identifies the set of values to which sums of rearrangements of an infinite series of vectors in Rn can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old.[1] In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a different method.[2]
In an expository article, Peter Rosenthal stated the theorem in the following way.[3]
- The set of all sums of rearrangements of a given series of vectors in a finite-dimensional real Euclidean space is either the empty set or a translate of a linear subspace (i.e., a set of the form v + M, where v is a given vector and M is a linear subspace).
See also
[edit]References
[edit]- ^ Lévy, Paul (1905), "Sur les séries semi-convergentes", Nouvelles Annales de Mathématiques, 64: 506–511.
- ^ Steinitz, Ernst (1913), "Bedingt Konvergente Reihen und Konvexe Systeme", Journal für die reine und angewandte Mathematik, 143: 128–175, doi:10.1515/crll.1913.143.128.
- ^ Rosenthal, Peter (April 1987), "The remarkable theorem of Lévy and Steinitz", American Mathematical Monthly, 94 (4): 342–351, doi:10.2307/2323094, JSTOR 2323094, MR 0883287.
- Banaszczyk, Wojciech (1991). Additive Subgroups of Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 1466. Berlin: Springer-Verlag. pp. 93–109. doi:10.1007/BFb0089147. ISBN 3-540-53917-4. MR 1119302. Zbl 0743.46002.
- Kadets, V. M.; Kadets, M. I. (1991). Rearrangements of series in Banach spaces. Translations of Mathematical Monographs. Vol. 86 (Translated by Harold H. McFaden from the Russian-language (Tartu) 1988 ed.). Providence, RI: American Mathematical Society. pp. iv+123. ISBN 0-8218-4546-2. MR 1108619.
- Kadets, Mikhail I.; Kadets, Vladimir M. (1997). Series in Banach spaces: Conditional and unconditional convergence. Operator Theory: Advances and Applications. Vol. 94. Translated by Andrei Iacob from the Russian-language. Basel: Birkhäuser Verlag. pp. viii+156. ISBN 3-7643-5401-1. MR 1442255.