Spaces of Analytic Functions

Abstract

In this chapter, we shall put a metric on the set of all analytic functions on a fixed region $G\subset \mathbb {C},$ and “compactness”, “converge”, “normality”, “uniform continuity”, and “equicontinuity” in this metric space is discussed. We shall also  discuss Hurwitz’s theorem, Montel’s theorem and  among the applications obtained is a proof of the Riemann mapping theorem.

In most sciences one generation tears down what

another has built and what one has established

another undoes. In mathematics alone each gene-

rations adds a new story to the old structure

Hermann Hanke