Abstract:
Very few we find in literature the order properties that product spaces of
ordered linear spaces and ordered normed spaces carry from their component
spaces. Some authors have studied certain order properties of ordered linear
spaces which are carried by their quotient spaces. A little work is done in
connection with the order properties of ordered normed spaces that the quotient
spaces inherit from their parent spaces.
In this thesis we have studied several order properties of ordered linear
spaces and ordered normed spaces that are preserved with or without conditions in
the formation of products. We have also studied several order properties of
quotient spaces of ordered linear spaces and ordered normed spaces by discussing
inheritance of order properties of the parent space X to its quotient space X/E.
This thesis consists of six chapters. Chapter 1 includes some basic
definitions and results. Chapter 2 provides almost all possible results available in
the literature regarding the topological properties that are preserved in the
formation of products. Chapter 3 is about the quotient spaces and includes several
results available in the literature for the quotient spaces of (i) topological spaces
(ii) linear spaces (iii) topological groups and (iv) topological linear spaces.
Chapter 4 and Chapter 5 include the research work. Six research papers
have been published from the work contained in these chapters; out of which
four are published in the journals approved by the Higher Education
Commission (HEC). Two more research papers are being submitted in some
journals of international repute. Chapter six also includes the research work
and presents only results from a past publication.
