Notion of Exhaustiveness and Generalized Ascoli Theorem

Abstract

The present dissertation entitled “THE NOTION OF EXHAUSTIVENESS AND GENERALIZED ASCOLI THEOREM” embodies a study carried out by me under the supervision of Dr. S.S. Bhatia, Professor and Head, School of Mathematics and Computer Applications, Thapar University, Patiala. In this dissertation, we study the notion of exhaustiveness which applies for both families and sequence of functions. This new notion is close to equicontinuity and describes the relation between pointwise convergence for functions and -convergence (continuous convergence). Using these results we have studied the Generalized Ascoli theorem dealing with exhuastiveness instead of equicontinuity. The work presented in this dissertation is divided into four chapters. Chapter I is introductory. In this chapter, we have represented some basic notations and definitions used in the sequel.In Chapter II we have studied the notion of exhaustiveness introduced by V. Gregoriades and N. Papanastassiou [5] which is close to equicontinuity. The notion of exhaustiveness enables us to view the convergence of a sequence of functions in terms of properties of the sequence and not of properties of functions as single members. In chapter III we have studied the notion of -convergence (known as continuous convergence) introduced by R. Das [4] which turned out to be useful for characterizing compactness in metric spaces. The notion of - convergence is stronger than pointwise convergence. On the other hand, if the limit function f is continuous, then this convergence is weaker than uniform convergence. Chapter IV is devoted to the study of the Generalization of Ascoli Theorem using the notion of exhaustiveness. Towards the end, references of various publications cited in the present dissertation have been reported.