In mathematics, the Lévy–Steinitz theorem identifies the set of values to which rearrangements of an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
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
Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician.
Biography
Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...
filled in a gap in Lévy's proof and also proved the result by a different method.
In an expository article,
Peter Rosenthal
Peter Michael Rosenthal (born June 1, 1941) is Canadian-American Professor Emeritus of Mathematics at the University of Toronto, an adjunct professor of Law at the University of Toronto, and a lawyer in private practice.
Early life
Rosenthal g ...
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 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
Mathematical series
Permutations
Summability theory
Theorems in analysis