• Corpus ID: 15903573

Topologies on the space of holomorphic functions

@article{Krantz2007TopologiesOT,
  title={Topologies on the space of holomorphic functions},
  author={Steven G. Krantz},
  journal={arXiv: Complex Variables},
  year={2007}
}
  • S. Krantz
  • Published 12 July 2007
  • Mathematics
  • arXiv: Complex Variables
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. 
Frechet topologies on hypoelliptic sheaves
In [2] it is shown that on the space of holomorphic functions convergence with respect to any ‘reasonable’ Fréchet topology implies uniform convergence on compact sets. The question is raised whether
Non-natural topologies on spaces of holomorphic functions
It is shown that every proper Frechet space with weak*-separable dual admits uncountably many inequivalent Frechet topologies. This applies, in particular, to spaces of holomorphic functions so

References

Applied Functional Analysis: Main Principles and Their Applications
1 The Hahn-Banach Theorem Optimization Problems.- 1.1 The Hahn-Banach Theorem.- 1.2 Applications to the Separation of Convex Sets.- 1.3 The Dual Space C[a,b]*.- 1.4 Applications to the Moment