ABSTRACT
A number of results about holomorphic functions can be expressed very conveniently in terms of a topology on the vector space O D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq831.tif"/> of holomorphic functions in an open subset D ⊆ ℂ n , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq832.tif"/> and indeed it is but natural in analysis to study various topological vector spaces. It was observed in section A that if a sequence of functions in O D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq833.tif"/> converges uniformly on compact subsets of D, then the limit function also belongs to O D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq834.tif"/> ; that suggests imposing on O D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq835.tif"/> the topology for which this is the appropriate notion of convergence, the compact-open topology.
