Lévy–Steinitz theorem explained

In mathematics, the Lévy–Steinitz theorem identifies the set of values to which 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

• Riemann series theorem

References

• Book: Banaszczyk, Wojciech . Additive Subgroups of Topological Vector Spaces . . 1466 . . Berlin . 1991 . 93–109 . 3-540-53917-4 . 1119302. 0743.46002. 10.1007/BFb0089147.
• Book: Kadets . V. M. . Kadets . M. I. . Mikhail Kadets . Rearrangements of series in Banach spaces . Translated by Harold H. McFaden from the Russian-language (Tartu) 1988 . Translations of Mathematical Monographs . 86 . American Mathematical Society . Providence, RI . 1991 . iv+123 . 0-8218-4546-2 . 1108619.
• Book: Kadets . Mikhail I. . Kadets . Vladimir M. . Series in Banach spaces: Conditional and unconditional convergence . Translated by Andrei Iacob from the Russian-language . Operator Theory: Advances and Applications . 94 . Birkhäuser Verlag . Basel . 1997 . viii+156 . 3-7643-5401-1 . 1442255.

Notes and References

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