Exploring Extreme Points and Related Properties of Tsirelson Space (thesis)

Abstract

Tsirelson space was constructed in 1974 as the first example of a Banach space without an embedded c0 or lp space. In 1989, Casazza and Shura wrote a book Tsirelson's Space devoted to Tsirelson space and its many properties. In this thesis, we give two representations of Tsirelson space and give an exposition of many results found in the Casazza-Shura book. In the final chapters, we make two mathematical contributions. First, we give new examples of extreme points of the unit ball of Tsirelson space, which expands the list of known ones from the Casazza-Shura book. Secondly, we improve the bounds on j(n), which roughly measures the complexity of norming a vector of length n. We give an O(log2(n)) lower bound and an O( p n) upper bound. Both of these results answer questions from Tsirelson's Space, the second of which improves upon the O(n) upper bound given by Casazza and Shura. This thesis is written to be accessible to mathematicians outside of functional analysis.