Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on ℋ ( U )

Sean Dineen

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 1, page 19-54
  • ISSN: 0373-0956

Abstract

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This article is devoted to a study of locally convex topologies on H ( U ) (where U is an open subset of the locally convex topological vector space E and H ( U ) is the set of all complex valued holomorphic functions on E ). We discuss the following topologies on H ( U ) :(a) the compact open topology I 0 ,(b) the bornological topology associated with I 0 ,(c) the ported topology of Nachbin I ω ,(d) the bornological topology associated with I ω  ; and(e) the I ω topological of Nachbin.For U balanced we show these topologies are related to various kinds of convergence of the Taylor series at o . This in turn allows us to obtain criterion for equality of different topologies on H ( U ) and to construct counterexamples when we do not have equality.

How to cite

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Dineen, Sean. "Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on ${\mathcal {H}} (U) $." Annales de l'institut Fourier 23.1 (1973): 19-54. <http://eudml.org/doc/74110>.

@article{Dineen1973,
abstract = {This article is devoted to a study of locally convex topologies on $\{\bf H\}(U)$ (where $U$ is an open subset of the locally convex topological vector space $E$ and $\{\bf H\}(U)$ is the set of all complex valued holomorphic functions on $E$). We discuss the following topologies on $\{\bf H\}(U)$:(a) the compact open topology $\{\bf I\}_0$,(b) the bornological topology associated with $\{\bf I\}_0$,(c) the ported topology of Nachbin $\{\bf I\}_\omega $,(d) the bornological topology associated with $\{\bf I\}_\omega $ ; and(e) the $\{\bf I\}_\omega $ topological of Nachbin.For $U$ balanced we show these topologies are related to various kinds of convergence of the Taylor series at $o$. This in turn allows us to obtain criterion for equality of different topologies on $\{\bf H\}(U)$ and to construct counterexamples when we do not have equality.},
author = {Dineen, Sean},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {19-54},
publisher = {Association des Annales de l'Institut Fourier},
title = {Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on $\{\mathcal \{H\}\} (U) $},
url = {http://eudml.org/doc/74110},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Dineen, Sean
TI - Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on ${\mathcal {H}} (U) $
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 1
SP - 19
EP - 54
AB - This article is devoted to a study of locally convex topologies on ${\bf H}(U)$ (where $U$ is an open subset of the locally convex topological vector space $E$ and ${\bf H}(U)$ is the set of all complex valued holomorphic functions on $E$). We discuss the following topologies on ${\bf H}(U)$:(a) the compact open topology ${\bf I}_0$,(b) the bornological topology associated with ${\bf I}_0$,(c) the ported topology of Nachbin ${\bf I}_\omega $,(d) the bornological topology associated with ${\bf I}_\omega $ ; and(e) the ${\bf I}_\omega $ topological of Nachbin.For $U$ balanced we show these topologies are related to various kinds of convergence of the Taylor series at $o$. This in turn allows us to obtain criterion for equality of different topologies on ${\bf H}(U)$ and to construct counterexamples when we do not have equality.
LA - eng
UR - http://eudml.org/doc/74110
ER -

References

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