From: "Patrick J. Fitzsimmons" <pfitzsim@ucsd.edu>
Subject: Re: Sums of permutations of infinite series
Date: 7 Jul 1999 11:00:02 -0500
Newsgroups: sci.math.research,sci.math
Keywords: Levy-Steinitz theorem

In article <slrn7o1vsg.rep.spam@dad.dad.org>, Do not Spam
<spam@dad.org> wrote:

> Associate with every infinite series, S, its sum set, by which I mean the set
> of
> sums of permutations of S, which have finite sums.  By a permutation I mean a
> one-to-one map of a set onto itself.
> 
> It is easy to see that the possible sum sets of infinite series of real
> numbers
> are either empty, single points, or the whole real line.
> 
> What can you say about sum sets of series of elements of Euclidean n-space?
> 
> Jack Feldman conjectures that the possible sum sets are either empty or linear
> manifolds.

That this is indeed the case is a result known as the L\'evy-Steinitz
Theorem.  A nice exposition of the theorem, by P. Rosenthal, can be
found in the American Math. Monthly [vol. 94 (1987) 342--351].


--P.F.
==============================================================================
[Citation from MathSciNet: -- djr]
   88d:40005 40A05
   Rosenthal, Peter(3-TRNT)
   The remarkable theorem of Levy and Steinitz. 
   Amer. Math. Monthly 94 (1987), no. 4, 342--351.

   Consider a given real series. Then it is familiar that the set of all
   sums of convergent rearrangements of the series is either empty or a
   single point on the set of all reals. For a complex series, the
   corresponding set is either empty or a single point on the set of all
   points of a line in the whole complex plane. More generally, there is
   a natural extension to series of vectors in a finite-dimensional real
   Euclidean space. Although this result goes back a long time, it is not
   nearly so familiar as the result for a real series. As the author says
   in his introductory remarks, "the purpose of this article is to make
   this beautiful result more widely known".

                           Reviewed by B. Kuttner