On Levy-Steinitz Theorem (LS)

Abstract

The over 150 years old Riemann's rearrangement-of-series Theorem (R)
states that one can rearrange the terms of a given conditionally
convergent series of real numbers to obtain a "rearranged" series that
converges to an arbitrarily given real.

LS (early 20th century) is a beautiful generalization of R.

It tells us that in R^{n}, *SS(h)*, the set-of-sums
of the convergent rearrangements of a given series *Σh(n)*, where
*h = (h(1), h(2),.,h(n),.)* is a given sequence in R^{n},
if nonempty, is an affine subspace (a displacement of a linear subspace) of R^{n}.
R, LS and the situation beyond finite dimensions will be discussed. An interesting
instance of *h* in the infinite dimensional setting that extends
Riemann's theorem R and Levy-Steinitz Theorem LS will be presented.