2 3 Ju l 2 00 7 Topologies on the Space of Holomorphic Functions ( REVISED ) Steven

We remark that the result presented here can be proved more simply using the closed graph theorem. However we believe that our results can be used to prove a more interesting result. Details to follow in a later paper. One of the remarkable features of the space of holomorphic functions (in either one or several variables) is that the standard Frechet space topologies— say, for example, the L or Bergman norm—control a stronger (and simpler) and more useful topology, namely uniform convergence on compact sets. This simple fact lies at the heart of many key results in basic complex function theory—for example the completeness of many important function spaces. It is natural to wonder whether this property is universal. Is it the case that any Frechet space topology on the space of holomorphic functions implies uniform convergence on compact sets (equivalently, convergence in the compact-open topology)? The surprising answer to this question—suitably formulated—is “yes”, and that is the main result of the present paper. It is a pleasure to thank Peter Pflug for posing the question that led to this paper, and for early discussions of the matter.