Compactness and Convergence in the Space Of Analytic Functions

Abstract

Here, our main aim is to study compactness and convergence in the space of continuous functions defined on a fixed region G subset of complex plane. First, we will define an appropriate metric in which we will study compactness and convergence. For defining compactness we will introduce the concept of normal set and then we will prove that normal closure is compact. Subsequently, we will prove a variant of a famous theorem i.e. Arzela-Ascoli theorem. Then we will divert our attention in studying compactness and convergence in the space of analytic functions defined on a fixed region G. The analytic functions are having an exceptional importance as this class is sufficiently large. It includes the majority of functions which are encountered in the principal problems of mathematics and applications to science and technology. Here, in our discussion we visualize these analytic functions as points in a metric space. Also, here we have proved Hurwitz and Montel theorem. In the last section of this dissertation we have studied the space of meromorphic functions defined in a fixed region G.