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