Convergence in ordered vector spaces

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Thesis Type: Postgraduate

Institution Of The Thesis: Gazi University, Fen Bilimleri Enstitüsü, Turkey

Approval Date: 2024

Thesis Language: Turkish

Student: Büşra ÇELİK

Supervisor: Cüneyt Çevik

Open Archive Collection: AVESIS Open Access Collection

Abstract:

In ordered vector spaces, there are several natural ways to describe convergence using only ordering. These are commonly known as order convergences. Order convergence of nets is widely used in related studies. For example, order convergence is used for order continuous norms when studying normed Riesz spaces. At the same time, order convergence is used to define order continuous operators, which are continuous concerning this convergence for operators between Riesz spaces. Commonly used convergence definitions for nets derive from the convergence descriptions given for sequences. However, one problem with this convergence is that it depends on the tail and the beginning of the net’s row convergence. In 2005, Abramovich and Sirotkin proposed a new and improved definition for the convergence of nets in Riesz space theory. Subsequently, this definition has been used in studies related to convergence in Riesz spaces. On the other hand, in ordered vector spaces, there is usually no norm that is as proper as the order unit norm. However, such a norm can be defined by a similar concept called relative uniform convergence. In this study, in addition to these two order convergence definitions, order convergences, which have been discussed in recent studies and which expand the results according to the structure of the space, are also examined.

Key Words : Riesz space, ordered vector space, convergence