On the Lévy-Steinitz Theorem

Abstract

This thesis is a further study of Riemann's Theorem of rearrangements of series. The theorem states: (1) If $\sum a_j$ is a conditionally convergent series of real numbers and a is a real number, then there is a rearrangement of the series which converges to a. (2) Any rearrangement of an absolutely convergent series in $\mathbb{R}^n$ converges to the same element. The focus of this thesis is to cover the $\mathbb{R}^n$ generalization of Riemann's Theorem and to look at some counter examples in infinite dimensional spaces. Chapter 1 gives the proofs to Riemann's Theorem. Chapter 2 covers the Lévy-Steinitz theorem, which states that the set of sums of convergent rearrangements of a given series is the translate of a subspace of $\mathbb{R}^n$ In chapter 3, a sufficient condition is given for the sum range to be the whole space $\mathbb{R}^n$ The discussion in chapter 4 provides some counter examples proving that in general there is no Levy-Steinitz theorem in the space $L^2$(0,1) and for certain $\downharpoonright^p$ spaces. Also given is an example of a series which has sum range equal to $L^2$. Riemann's Theorem is a well known result in analysis that can be found in many calculus textbooks. Two good references for this result are [4],[5]. The results described in chapters 2 through 4 can be found in [1],[2]. For more recent results related to the Lévy-Steinitz theorem, including results in infinite-dimensional spaces, see [3].