Abstract
For any weighted space HV(G) of holomorphic functions on an open set G ⊂ ₵Nwith a topology stronger than that of uniform convergence on the compact sets and for any quasibarrelled space E we prove the topological isomorphism \(HV(G,E_{b}^{\prime})={\cal L}_b(E,HV(G))\)and derive a similar, more complicated isomorphism for weighted spaces of continuous functions. This generalizes results of [3], [7] and [6] and should be compared with the ∈-product representations for the corresponding spaces of functions with o-growth conditions. At the end we also show the topological isomorphism HV1(G1, HV2(G2)) = H (V1 ⊗ V2)(G1 × G2).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K.-D. Bierstedt, The approximation property for weighted function spaces; Tensor products of weighted spaces, pp. 3-25, 26-58 in Function Spaces and Dense Approximation (Proc. Conference Bonn 1974), Bonner Math. Schriften 81 (1975).
K.-D. Bierstedt, The approximation-theoretic localization of Schwartz’s approximation property for weighted locally convex function spaces and some examples, pp. 93-149 in Functional Analysis, Holomorphy, and Approximation Theory (Proc. Conference Rio de Janeiro 1978), Springer Lecture Notes in Math. 843 (1981).
K.D. Bierstedt, J. Bonet, Dual density conditions in (DF)-spaces, I., Results Math. 14 (1988), 241-274; IL, Bull. Soc. Roy. Sci. Liège 57 (1988), 567–589.
K.D. Bierstedt, J. Bonet, Biduality in Fréchet and (LB)-spaces, pp. 113-133 in Progress in Functional Analysis (Proc. Conference Peñíscola 1990), North-Holland Mathematics Studies 170 (1992).
KD Bierstedt R Meise (1986) ArticleTitleDistinguished echelon spaces and the projective description of weighted inductive limits of type\({\cal V}_{d}C(X)\) Aspects of Mathematics and its Applications (dedicated to L. Nachbin) 34 169–226 Occurrence Handle849552 Occurrence Handle10.1016/S0924-6509(09)70255-7
K.D. Bierstedt, J. Bonet, A. Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), 271–297.
K.D. Bierstedt, J. Bonet, J. Schmets, (DF)-spaces of type CB(X, E) and \(C{\overline V}(X,E)\), Note Mat. 10, Suppl. no. 1 (dedicated to the memory of G. Köthe) (1990), 127–148.
K.-D. Bierstedt, B. Gramsch, R. Meise, Approximationseigenschaft, Lifting und Kohomologie bei lokalkonvexen Produktgarben, Manuscripta Math. 19 (1976), 319–364.
K.D. Bierstedt, R. Meise, W.H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107–160.
W. Cutrer, Vitushkin’s localization operator and holomorphic functions on polydiscs, Duke Math. J. 39 (1972), 737–744.
S. Dierolf, On spaces of continuous linear mappings between locally convex spaces, Habilitationsschrift (Univ. München 1983), Note Mat. 5 (1985), 147–255.
K.-G. Große-Erdmann, The Borel-Okada theorem revisited, Habilitationsschrift (Fern-Univ. Hagen 1992).
A. Grothendieck, Sur certains espaces de fonctions holomorphes I, J. Reine Angew. Math. 192 (1953), 35–64.
A. Grothendieck, Espaces vectoriels topologiques, São Paulo (1954).
H. Jarchow, Locally Convex Spaces, Mathematische Leitfäden, B.G. Teubner (1981).
J.L. Kelley, General Topology, van Nostrand (1955, reprint 1969).
L. Nachbin, Elements of Approximation Theory, van Nostrand Mathematical Studies 14 (1967).
A. Peris, Quasinormable spaces and the problem of topologies of Grothendieck, Ann. Acad. Sci. Fenn., Ser. A.L Math. 19 (1994), 167–203.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bierstedt, K.D., Holtmanns, S. An operator representation for weighted spaces of vector valued holomorphic functions. Results. Math. 36, 9–20 (1999). https://doi.org/10.1007/BF03322097
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322097