Abstract
If E is a complex (DFC)-space (see § 2), we show that E leads to pure uniform holomorphy (see §2) if and only if its Fréchet dual space E′ is separable (see Theorem 1, where these two conditions have other eight equivalent ones). By using a theorem of Mujica (see §4), we consider the (DFC)-space K(K) of germs around K of holomorphic C-valued functions where K is a nonvoid compact subset of a complex metrizable locally convex space E, and ℋ(K) is endowed with the topology ℐ0 obtained as an inductive limit of compact-open topologies (see §4). Not only Theorem 1 applies to ℋ(K), with E replaced by ℋ(K) in its statement, but also ℋ(K) leads to pure uniform holomorphy if and only if E is separable (see Theorem 2).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.-F. Colombeau, Differential Calculus and Holomorphy, North-Holland (1982), Netherlands.
S. Dineen, Complex Analysis in locally convex spaces, North-Holland (1981), Netherlands.
A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasiliensis Mathematicae 3 (1954), 57–123.
R. Hollstein, (DCF)-Räume und lokalkonvexe Tensorprodukte, Archiv der Mathematik 29 (1977), 524–531.
R. Hollstein, Tensorprodukte von stetigen linearen Abbildungen in (F) und (DCF)-Räumen, Journal für die reine und angewandte Mathematik 301 (1978), 191–204.
J. Horvath, Topological vector spaces and distributions, Addison-Wesley (1966), USA.
J. Mujica, Domains of holomorphy in (DFC)-spaces, Functional Analysis, Holomorphy, and Approximation Theory (Ed.: S. Machado), Lecture Notes in Mathematics 843 (1981), 500–533.
J.Mujica, A new topology on the space of germs of holomorphic functions, IMECC, Universidade Estadual de Campinas, Relatório Interno 193 (1981), Brazil.
J. Mujica, Holomorphic approximation in pseudoconvex Riemann domains over Fréchet spaces with basis, IMECC, Universidade Estadual de Campinas, Relatório Interno 237 (1983), Brazil.
L. Nachbin, Uniformité d’holomorphie et type exponentiel, Séminaire Pierre Lelong, Lecture Notes in Mathematics 205 (1971), 216–224.
L. Nachbin, Recent developments in infinite dimensional holomorphy, Bulletin of the American Mathematical Society 79 (1973), 625–640.
L. Nachbin, A glimpse at infinite dimensional holomorphy, Proceedings on Infinite Dimensional Holomorphy (Eds.: T. L. Hayden and T. J. Suffridge), Lecture Notes in Mathematics 364 (1974), 69–79.
O. Nicodemi, Homomorphisms of algebras of germs of holomorphic functions, Functional Analysis, Holomorphy and Approximation Theory (Ed.: S. Machado), Lecture Notes in Mathematics 843 (1981), 534–546.
P. Noverraz, Pseudo-convexité, convexité polynomiale et domaines d’holomorphie en dimension infinie, North-Holland (1973), Netherlands.
Appendix to the References
J. Mujica, A Banach-Dieudonné theorem for germs of holomorphic mappings, Journal of Functional Analysis 57 (1984), 31–48.
J. Mujica, Holomorphic approximation in infinite dimensional Riemann domains, Studia Mathematica (to appear).
L. Nachbin, A glance at holomorphic factorization an uniform holomorphy, Complex Analysis, Functional Analysis and Approximation Theory (Ed.: J. Mujica), North-Holland (to appear), Netherlands.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nachbin, L. On pure uniform holomorphy in spaces of holomorphic germs. Results. Math. 8, 117–122 (1985). https://doi.org/10.1007/BF03322663
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322663