Topologies on the space of holomorphic functions

TL;DR

The paper investigates how various Frechet space topologies on holomorphic functions influence the topology of uniform convergence on compact sets, using the closed graph theorem for a simplified proof.

Contribution

It demonstrates that arbitrary Frechet topologies on holomorphic functions determine the uniform convergence topology, with potential for broader applications using the presented techniques.

Findings

Frechet space topologies control uniform convergence topology

Closed graph theorem simplifies the proof

Potential for more general results using these methods

Abstract

We show that a fairly arbitrary Frechet space topology on the space of holomorphic functions on a domain controls the topology of uniform convergence on compact sets. In fact it turns out that the result we present can be proved more simply using the closed graph theorem. However, we believe that the techniques presented here may be used to prove a more interesting result. Details to appear later.