Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Extension of holomorphic mappings from $ E$ to $ E''$


Author: Luiza A. Moraes
Journal: Proc. Amer. Math. Soc. 118 (1993), 455-461
MSC: Primary 46G20
DOI: https://doi.org/10.1090/S0002-9939-1993-1139471-0
MathSciNet review: 1139471
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Assuming that $ E$ is a distinguished locally convex space and $ F$ is a complete locally convex space, we prove that there exists an open subset $ V$ of $ E''$ that contains $ E$ and such that every holomorphic mapping $ f:E \to F$ whose restriction $ f\vert B$ is $ \sigma (E,E')$-uniformly continuous for every bounded subset $ B$ of $ E$ has a unique holomorphic extension $ \tilde f:V \to F$ such that $ \tilde f\vert B$ is $ \sigma (E'',E')$-uniformly continuous for every bounded subset $ B$ of $ V$. We show that in many cases we can take $ V = E''$. This is the case when $ E''$ is a locally convex space where every $ G$-holomorphic mapping that is bounded in a neighbourhood of the origin is locally bounded.


References [Enhancements On Off] (What's this?)

  • [1] R. Aron and P. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 3-24. MR 508947 (80e:46029)
  • [2] P. Boland, Holomorphic functions on nuclear spaces, Trans. Amer. Math. Soc. 209 (1975), 275-281. MR 0388094 (52:8931)
  • [3] S. Dineen, Holomorphically complete locally convex topological vector spaces, Séminaire Pierre Lelong 1971-72, Lecture Notes in Math., vol. 332, Springer, Berlin, 1973, pp. 77-111. MR 0377512 (51:13684)
  • [4] -, Complex analysis in locally convex spaces, North-Holland Math. Stud., vol. 57, North-Holland, Amsterdam, 1981. MR 640093 (84b:46050)
  • [5] J. Horváth, Topological vector spaces and distributions, vol. I, Addison-Wesley, Reading, MA, 1966. MR 0205028 (34:4863)
  • [6] J. Jarchow, Locally convex spaces, Teubner, Stuttgart, 1981. MR 632257 (83h:46008)
  • [7] R. Meise and D. Vogt, Counterexamples in holomorphic functions on nuclear Fréchet spaces, Math. Z. 182 (1983), 167-177. MR 689294 (84m:46048)
  • [8] L. A. Moraes, The Hahn-Banach extension theorem for some spaces of $ n$-homogeneous polynomials, Functional Analysis: Surveys and Recent Results III (K. D. Bierstedt and B. Fuchsteiner, eds.), North-Holland Math. Stud., vol. 90, North-Holland, Amsterdam, 1984, pp. 265-274. MR 761386 (86e:46038)
  • [9] -, A Hahn-Banach extension theorem for some holomorphic functions, Complex Analysis, Functional Analysis and Approximation Theory (J. Mujica, ed.), North-Holland Math. Stud., vol. 125, North-Holland, Amsterdam, 1986, pp. 205-220. MR 893417 (88f:46094)
  • [10] -, Quotients of spaces of holomorphic functions on Banach spaces, Proc. Roy. Irish Acad. Sect. A 87 (1987), 181-186. MR 941714 (89i:46051)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46G20

Retrieve articles in all journals with MSC: 46G20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1139471-0
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society