Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-29T19:44:56.228Z Has data issue: false hasContentIssue false
  • English
  • Français

The locally convex topology on the space of meromorphic functions

Part of: Entire and meromorphic functions, and related topics Topological fields Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Karl-Goswin Grosse-Erdmann
Karl-Goswin Grosse-Erdmann
Affiliation:
Fachbereich Mathematik FernuniversitätHagen Postfach 940 D-58084 Hagen, Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a positive answer to a question of Horst Tietz. A theorem of his that is related to the Mittag-Leffler theorem looks like a duality restult under some locally convex topology on the space of meromorphic functions. Tietz has posed the problem of finding such a topology. It is shown that a topology introduced by Holdguün in 1973 solves the problem. The main tool in the study of this topology is a projective description of it that is derived here. We also argue that Holdgrün's topology is the natural locally convex topology on the space of meromorphic functions.

MSC classification

Secondary: 30D30: Meromorphic functions, general theory 46E10: Topological linear spaces of continuous, differentiable or analytic functions 12J99: None of the above, but in this section 46E25: Rings and algebras of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 3 , December 1995 , pp. 287 - 303
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Arens, R., ‘Linear topological division algebras’, Bull. Amer. Math. Soc. 53 (1947), 623–630.CrossRefGoogle Scholar
[2]Barsukov, E. G., ‘Complete systems in the space of analytic functions with isolated singularities of unique character’, in: Mathematical analysis and its applications, vol. VI (Rostov State University, Rostov-on-Don, 1974) pp. 228–237 (in Russian).Google Scholar
[3]Barsukov, E. G., ‘Maximal ideals in an algebra of meromorphic functions’, in: Theory of functions. Differential equations and their applications. No. 1 (Kalmyk State University, Elista, 1976) pp. 23–34 (in Russian).Google Scholar
[4]Barsukov, E. G., ‘On abstract meromorphic algebras’, in: Mathematical analysis and its applications (Rostov State University, Rostov-on-Don, 1981) pp. 7–10 (in Russian).Google Scholar
[5]Barsukov, E. G. and Khaplanov, M. G., ‘A space of meromorphic functions’, Mat. Zametki 17 (1975), 589–598 (in Russian),Google Scholar
English translation: Math. Notes 17 (1975), 350–355.Google Scholar
[6]Beckenstein, E., Narici, L. and Suffel, C., Topological algebras (North-Holland, Amsterdam, 1977).Google Scholar
[7]Guerrero, P. Bobillo, ‘Topologies on M(A)’, in: Proceedings of the eleventh annual conference of Spanish mathematicians (Murcia, 1970) (Universidad Complutense de Madrid, Madrid, 1973) pp. 22–30 (in Spanish).Google Scholar
[8]Boehme, T. K., ‘On the limits of the Gelfand-Mazur theorem’, Proceedings of the conference on convergence spaces (Reno, 1976) (University of Nevada, Reno, 1976) pp. 1–4.Google Scholar
[9]Burmann, H.-W. and Holdgrün, H. S., ‘Ein vollständiger induktiver Limes aus gewissen meromorphen Funktionen’, Math. Z. 102 (1967), 89–109.CrossRefGoogle Scholar
[10]Cima, J. A. and Pfaltzgraff, J. A., ‘The Hornich topology for meromorphic functions in the disk’, J. Reine Angew. Math. 235 (1969), 207–220.Google Scholar
[11]Cima, J. and Schober, G., ‘On space of meromorphic functions’, Rocky Mountain J. Math. 9 (1979), 527–532.CrossRefGoogle Scholar
[12]Constantinescu, T. and Gheondea, A., ‘Algebraic aspects of the functional calculus for meromorphic functions’, Rev. Roumaine Math. Pures Appl. 27 (1982), 949–956.Google Scholar
[13]Conway, J. B., Functions of one complex variable, 2nd edition (Springer, New York, 1978).CrossRefGoogle Scholar
[14]Floret, K. and Wloka, J., Einführung in die Theorie der lokalkonvexen Räume (Springer, Berlin, 1968).CrossRefGoogle Scholar
[15]Golovin, V. D., ‘Duality in spaces of holomorphic functions with singularities’, Dokl. Akad. Nauk SSSR 168 (1966), 9–12 (in Russian).Google Scholar
English translation: Soviet Math. Dokl. 7 (1966), 571–574.Google Scholar
[16]Golovin, V. D., ‘On some spaces of holomorphic functions with isolated singularities’, Mat. Sb. 73 (115) (1967), 21–41 (in Russian).Google Scholar
English translation: Math. USSR - Sb. 2 (1967), 17–33.Google Scholar
[17]Holdgrün, H. S., ‘Fastautomorphe Funktionen auf komplexen Räumen’, Math. Ann. 203 (1973), 35–64.CrossRefGoogle Scholar
[18]Jarchow, H., Locally convex spaces (Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[19]Köthe, G., Topological vector spaces, 1 (Springer, Berlin, 1969).Google Scholar
[20]Kürschák, J., ‘Über Limesbildung und allgemeine Körpertheorie’, J. Reine Angew. Math. 142 (1913), 211–253.CrossRefGoogle Scholar
[21]Ostrowski, A., ‘Über Folgen analytischer Funktionen und einige Verschärfungen des Picardschen Statzes’, Math. Z. 24 (1926), 215–258.CrossRefGoogle Scholar
[22]Remmert, R., Theory of complex functions (Springer, New York, 1991).CrossRefGoogle Scholar
[23]Remmert, R., Funktionentheorie, II (Springer, Berlin, 1991).CrossRefGoogle Scholar
[24]Rudin, W., Real and complex analysis, 3rd edition (McGraw-Hill, New York, 1987).Google Scholar
[25]Tietz, H., ‘Zur Klassifizierung meromorpher Funktionen auf Riemannschen Flächen’, Math. Ann. 142 (1961), 441–449.CrossRefGoogle Scholar
[26]Tietz, H., Private communication.Google Scholar
[27]Vakher, F. S. and Ryndina, V. V., ‘Algebras of meromorphic functions’, in: Mathematical analysis and its applications (Rostov State University, Rostov-on-Don, 1981), pp. 23–33 (in Russian).Google Scholar
[28]Waelbroeck, L., Topological vector spaces and algebras (Springer, Berlin, 1971).Google Scholar
[29]Warner, S., Topological fields (North-Holland, Amsterdam, 1989).Google Scholar
[30]Wieslaw, W., Topological fields (Marcel Dekkar, New York, 1988).Google Scholar
[31]Williamson, J. H., ‘On topologising the field C(t)’, Proc. Amer. Math. Soc. 5 (1954), 729–734.Google Scholar
[32]Zakharyuta, V. P. and Matvienko, A. I., ‘On bases in spaces of meromorphic functions with given poles’, in: Mathematical analysis and its applications (Rostov State University, Rostov-on-Don, 1981) pp. 55–60 (in Russian).Google Scholar