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 In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
of vectors in R''n'' 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

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Riemann series theorem In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms ...


References

* * * {{DEFAULTSORT:Lévy-Steinitz theorem Series (mathematics) Permutations Summability theory Theorems in mathematical analysis