On the Lévy-Steinitz Theorem

Loading...
Thumbnail Image
Authors
Meyer, Mark
Advisors
Fridman, Buma L.
Issue Date
2021-12
Type
Thesis
Keywords
Research Projects
Organizational Units
Journal Issue
Citation
Abstract

This thesis is a further study of Riemann's Theorem of rearrangements of series. The theorem states: (1) If 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 converges to the same element. The focus of this thesis is to cover the 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 In chapter 3, a sufficient condition is given for the sum range to be the whole space The discussion in chapter 4 provides some counter examples proving that in general there is no Levy-Steinitz theorem in the space (0,1) and for certain spaces. Also given is an example of a series which has sum range equal to . 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].

Table of Contents
Description
Thesis (M.S.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics
Publisher
Wichita State University
Journal
Book Title
Series
PubMed ID
DOI
ISSN
EISSN