Preduals of spaces of vector-valued holomorphic functions

Boyd, Christopher

Czechoslovak Mathematical Journal, Tome 53 (2003) no. 2, pp. 365-376

Abstract (VO)

For $U$ a balanced open subset of a Fréchet space $E$ and $F$ a dual-Banach space we introduce the topology $\tau _\gamma $ on the space ${\mathcal H}(U,F)$ of holomorphic functions from $U$ into $F$. This topology allows us to construct a predual for $({\mathcal H}(U,F),\tau _\delta )$ which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions.

Détail

Classification : 46A04, 46A20, 46A25, 46A32, 46E40, 46G20, 46G25
Keywords: holomorphic functions; Fréchet spaces; preduals

Bibliographie

[1] R. Alencar: An application of Singer’s theorem to homogeneous polynomials. Contemp. Math. 144 (1993), 1–8. | DOI | MR | Zbl

[2] A.  Alencar and K. Floret: Weak continuity of multilinear mappings on Tsirelson’s space. Quaestiones Math. 21 (1998), 177–186. | DOI | MR

[3] J. M.  Ansemil and S. Ponte: Topologies associated with the compact open topology on  ${\mathcal H}(U)$. Proc. R. Ir. Acad. 82A (1982), 121–129. | MR

[4] R.  Aron and M.  Schottenloher: Compact holomorphic mappings on Banach spaces and the approximation property. J.  Funct. Anal. 21 (1976), 7–30. | DOI | MR

[5] P. Aviles and J. Mujica: Holomorphic germs and homogeneous polynomials on quasinormable metrizable spaces. Rend. Math. 6 (1977), 117–127. | MR

[6] K.-D.  Bierstedt: An introduction to locally convex inductive limits. Functional Analysis and its Applications, World Scientific, 1988, pp. 35–132. | MR | Zbl

[7] K.-D.  Bierstedt and J.  Bonet: Biduality in Fréchet and (LB)-spaces. Progress in Functional Analysis, North-Holland Math. Studies, Vol. 170, 1992, pp. 113–133. | MR

[8] K.-D.  Bierstedt and J.  Bonet: Density condition in Fréchet and (DF)-spaces. Rev. Matem. Univ. Complut. Madrid 2 (1989), 59–76. | MR

[9] K.-D. Bierstedt, J. Bonet and A. Peris: Vector-valued holomorphic germs on Fréchet Schwartz spaces. Proc. Royal Ir. Acad. 94A (1994), 3–46.

[10] K.-D.  Bierstedt and R.  Meise: Aspects of inductive limits in spaces of germs of holomorphic functions on locally convex spaces and applications to study of $({\mathcal H}(U),\tau _\omega )$. Advances in Holomorphy (J. A.  Barroso, ed.). North-Holland Math. Studies 33 (1979), 111–178. | DOI | MR

[11] J. Bonet: A question of Valdivia on quasinormable Fréchet spaces. Canad. Math. Bull. 33 (1991), 301–314. | MR | Zbl

[12] J.  Bonet, P. Domański and J. Mujica: Completeness of spaces of vector-valued holomorphic germs. Math. Scand. 75 (1995), 250–260. | MR

[13] C.  Boyd: Preduals of the space of holomorphic functions on a Fréchet space. Ph.D. Thesis, National University of Ireland (1992).

[14] C.  Boyd: Distinguished preduals of the space of holomorphic functions. Rev. Mat. Univ. Complut. Madrid 6 (1993), 221–232. | MR

[15] C. Boyd: Montel and reflexive preduals of the space of holomorphic functions. Studia Math. 107 (1993), 305–325. | DOI | MR

[16] C.  Boyd: Some topological properties of preduals of spaces of holomorphic functions. Proc. Royal Ir. Acad. 94A (1994), 167–178. | MR | Zbl

[17] C.  Boyd and A. Peris: A projective description of the Nachbin ported topology. J.  Math. Anal. Appl. 197 (1996), 635–657. | DOI | MR

[18] J. F. Colombeau and D.  Lazet: Sur les théoremes de Vitali et de Montel en dimension infinie. C.R.A.Sc., Paris 274 (1972), 185–187. | MR

[19] H. Collins and W.  Ruess: Duals of spaces of compact operators. Studia Math. 74 (1982), 213–245. | DOI | MR

[20] A. Defant and W.  Govaerts: Tensor products and spaces of vector-valued continuous functions. Manuscripta Math. 55 (1986), 432–449. | MR

[21] S. Dineen: Complex Analysis on Locally Convex Spaces. North-Holland Math. Studies, Vol. 57, 1981. | MR

[22] S. Dineen: Holomorphic functions and Banach-nuclear decompositions of Fréchet spaces. Studia Math. 113 (1995), 43–54. | DOI | MR | Zbl

[23] K. G. Große-Erdmann: The Borel-Okada theorem revisited. Habilitation Schrift, 1992.

[24] R. B.  Holmes: Geometric Functional Analysis and its Applications. Springer-Verlag Graduate Texts in Math., Vol. 24, 1975. | DOI | MR | Zbl

[25] J.  Horváth: Topological Vector Spaces and Distributions, Vol. 1. Addison-Wesley, Massachusetts, 1966. | MR

[26] H.  Jarchow: Locally Convex Spaces. B. G.  Teubner, Stuttgart, 1981. | MR | Zbl

[27] N.  Kalton: Schauder decompositions in locally convex spaces. Cambridge Phil. Soc. 68 (1970), 377–392. | DOI | MR | Zbl

[28] J.  Mujica: A Banach-Dieudonné Theorem for the space of germs of holomorphic functions. J.  Funct. Anal. 57 (1984), 32–48. | DOI | MR

[29] J.  Mujica and L.  Nachbin: Linearization of holomorphic mappings on locally convex spaces. J.  Math. Pures Appl. 71 (1992), 543–560. | MR

[30] L.  Nachbin: Weak holomorphy. Unpublished manuscript.

[31] P. Pérez Carreras and J. Bonet: Barrelled Locally Convex Spaces. North Holland Math. Studies, Vol.  131. North Holland, Amsterdam, 1987. | MR

[32] A.  Peris: Quasinormable spaces and the problem of topologies of Grothendieck. Ann. Acad. Sci. Fenn. 19 (1994), 167–203. | MR | Zbl

[33] A.  Peris: Topological tensor products of a Fréchet Schwartz space and a Banach space. Studia Math. 106 (1993), 189–196. | DOI | MR | Zbl

[34] W. Ruess: Compactness and collective compactness in spaces of compact operators. J.  Math. Anal. Appl. 84 (1981), 400–417. | DOI | MR | Zbl

[35] R. A.  Ryan: Applications of topological tensor products to infinite dimensional holomorphy. Ph.D. Thesis, Trinity College, Dublin, 1980.

[36] H. H. Schaefer: Topological Vector Spaces. Third Printing corrected, Springer-Verlag, 1971. | MR | Zbl

[37] J.  Schmets: Espaces de fonctions continues. Lecture Notes in Math., Vol.  519, Springer-Verlag, Berlin-Heidelberg-New York, 1976. | MR | Zbl