ArticleOriginal scientific text

Title

Spaces of holomorphic mappings on Banach spaces with a Schauder basis

Authors 1

Affiliations

  1. Instituto de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13081-970 Campinas, SP, Brazil

Abstract

We show that if U is a balanced open subset of a separable Banach space with the bounded approximation property, then the space ℋ(U) of all holomorphic functions on U, with the Nachbin compact-ported topology, is always bornological.

Bibliography

  1. R. Aron, L. A. Moraes and R. Ryan, Factorization of holomorphic mappings in infinite dimensions, Math. Ann. 277 (1987), 617-628.
  2. S. B. Chae, Holomorphic germs on Banach spaces, Ann. Inst. Fourier (Grenoble) 21 (3) (1971), 107-141.
  3. G. Coeuré, Fonctions plurisousharmoniques sur les espaces vectoriels topologiques et applications à l'étude des fonctions analytiques, ibid. 20 (1) (1970), 361-432.
  4. G. Coeuré, Fonctionnelles analytiques sur certains espaces de Banach, ibid. 21 (2) (1971), 15-21.
  5. S. Dineen, The Cartan-Thullen theorem for Banach spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24 (1970), 667-676.
  6. S. Dineen, Holomorphy types on a Banach space, Studia Math. 39 (1971), 241-288.
  7. S. Dineen, Bounding subsets of a Banach space, Math. Ann. 192 (1971), 61-70.
  8. S. Dineen, Holomorphic functions on (c0,Xb)-modules, ibid. 196 (1972), 106-116.
  9. S. Dineen, Complex Analysis in Locally Convex Spaces, North-Holland Math. Stud. 57, North-Holland, Amsterdam, 1981.
  10. L. Gruman et C. Kiselman, Le problème de Levi dans les espaces de Banach à base, C. R. Acad. Sci. Paris 274 (1972), 1296-1299.
  11. Y. Hervier, Sur le problème de Levi pour les espaces étalés banachiques, ibid. 275 (1972), 821-824.
  12. W. Johnson, H. Rosenthal and M. Zippin, On bases, finite-dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506.
  13. B. Josefson, A counterexample in the Levi problem, in: Proceedings on Infinite Dimensional Holomorphy, T. Hayden and T. Suffridge (eds.), Lecture Notes in Math. 364, Springer, Berlin, 1974, 168-177.
  14. Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.
  15. J. Mujica, Holomorphic approximation in Fréchet spaces with basis, J. London Math. Soc. 29 (1984), 113-126.
  16. J. Mujica, Holomorphic approximation in infinite-dimensional Riemann domains, Studia Math. 82 (1985), 107-134.
  17. J. Mujica, Complex Analysis in Banach Spaces, North-Holland Math. Stud. 120, North-Holland, Amsterdam, 1986.
  18. L. Nachbin, On the topology of the space of all holomorphic functions on a given open subset, Indag. Math. 29 (1967), 366-368.
  19. L. Nachbin, Concerning spaces of holomorphic mappings, lecture notes, Rutgers Univ., New Brunswick, N.J., 1970.
  20. L. Nachbin, Sur les espaces vectoriels topologiques d'applications continues, C. R. Acad. Sci. Paris 271 (1970), 596-598.
  21. P. Noverraz, Pseudo-convexité, convexité polynomiale et domaines d'holomorphie en dimension infinie, North-Holland Math. Stud. 3, North-Holland, Amsterdam, 1973.
  22. P. Noverraz, Approximation of holomorphic or plurisubharmonic functions in certain Banach spaces, in: Proceedings on Infinite Dimensional Holomorphy, T. Hayden and T. Suffridge (eds.), Lecture Notes in Math. 364, Springer, Berlin, 1974, 178-185.
  23. A. Pełczyński, On the impossibility of embedding of the space L in certain Banach spaces, Colloq. Math. 8 (1961), 199-203.
  24. A. Pełczyński, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math. 40 (1971), 239-243.
  25. M. Schottenloher, The Levi problem for domains spread over locally convex spaces with a finite-dimensional Schauder decomposition, Ann. Inst. Fourier (Grenoble) 26 (4) (1976), 207-237.
Pages:
139-151
Main language of publication
English
Received
1995-10-26
Published
1997
Exact and natural sciences