In mathematics, the Lévy–Steinitz theorem identifies the set of values to which sums of rearrangements of an infinite series of vectors in R can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old. In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a different method.
In an expository article, Peter Rosenthal stated the theorem in the following way.
- 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
References
- 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.