|
Organizers |
Spaces of vector-valued meromorphic functions
by
Enrique Jordá
UPV
The aim of this work is to study the space M(\Omega, E) of meromorphic functions from a complex domain, \Omega, into a locally convex space E.
We discuss two notions of meromorphy that appear in [2], namely weak meromorphy and (strong) meromorphy. In case E is a locally complete space, we characterize those spaces in which both definitions of meromorphy coincide. We also study different locally convex topologies defined on M(\Omega, E) for a quasicomplete locally convex space E. Two of them are natural extensions of the topologies introduced by Grosse-Erdmann [1] in M(\Omega): The Mittag Leffler topology \tauML, defined by a locally convex projective limit, and the Holdgrün topology \tauHol, defined as a locally convex inductive limit. In [1] it is proved that these two topologies coincide in M(\Omega). We extend this result to M(\Omega, E) if E a Fréchet space. We show that in general \tauHol is finer than \tauML. Moreover we prove that every step in the inductive limit \tauHol is topologically isomorphic to the space of vector valued holomorphic functions H(\Omega, E). If E is Fréchet, we characterize those subspaces of M(\Omega, E) which are a Fréchet space or a webbed space. The relation with the \epsilon product of Schwartz M(\Omega)\epsilonE is also analyzed.
References
[1] Grosse-Erdmann, K-G.
The locally convex topology on the space of
meromorphic functions. J. Austral. Math. Soc. (Series A)
59 (1995) 287-303.
[2] Grosse Erdmann, K-G. The Borel Okada
theorem revisited. Habilitation Fernuniv. Hagen, 1993.
We report on research done under the advice of José Bonet and Manuel Maestre.
(P)
Date received: April 5, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caei-70.