| Title:
|
Preduals of spaces of vector-valued holomorphic functions (English) |
| Author:
|
Boyd, Christopher |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
53 |
| Issue:
|
2 |
| Year:
|
2003 |
| Pages:
|
365-376 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
holomorphic functions |
| Keyword:
|
Fréchet spaces |
| Keyword:
|
preduals |
| MSC:
|
46A04 |
| MSC:
|
46A20 |
| MSC:
|
46A25 |
| MSC:
|
46A32 |
| MSC:
|
46E40 |
| MSC:
|
46G20 |
| MSC:
|
46G25 |
| idZBL:
|
Zbl 1028.46063 |
| idMR:
|
MR1983458 |
| . |
| Date available:
|
2009-09-24T11:02:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127806 |
| . |
| Reference:
|
[1] R. Alencar: An application of Singer’s theorem to homogeneous polynomials.Contemp. Math. 144 (1993), 1–8. Zbl 0795.46033, MR 1209441, 10.1090/conm/144/1209441 |
| Reference:
|
[2] A. Alencar and K. Floret: Weak continuity of multilinear mappings on Tsirelson’s space.Quaestiones Math. 21 (1998), 177–186. MR 1701813, 10.1080/16073606.1998.9632038 |
| Reference:
|
[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 0669471 |
| Reference:
|
[4] R. Aron and M. Schottenloher: Compact holomorphic mappings on Banach spaces and the approximation property.J. Funct. Anal. 21 (1976), 7–30. MR 0402504, 10.1016/0022-1236(76)90026-4 |
| Reference:
|
[5] P. Aviles and J. Mujica: Holomorphic germs and homogeneous polynomials on quasinormable metrizable spaces.Rend. Math. 6 (1977), 117–127. MR 0458170 |
| Reference:
|
[6] K.-D. Bierstedt: An introduction to locally convex inductive limits.Functional Analysis and its Applications, World Scientific, 1988, pp. 35–132. Zbl 0786.46001, MR 0979516 |
| Reference:
|
[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 1150741 |
| Reference:
|
[8] K.-D. Bierstedt and J. Bonet: Density condition in Fréchet and (DF)-spaces.Rev. Matem. Univ. Complut. Madrid 2 (1989), 59–76. MR 1057209 |
| Reference:
|
[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. |
| Reference:
|
[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. MR 0520658, 10.1016/S0304-0208(08)70757-6 |
| Reference:
|
[11] J. Bonet: A question of Valdivia on quasinormable Fréchet spaces.Canad. Math. Bull. 33 (1991), 301–314. Zbl 0698.46002, MR 1127750 |
| Reference:
|
[12] J. Bonet, P. Domański and J. Mujica: Completeness of spaces of vector-valued holomorphic germs.Math. Scand. 75 (1995), 250–260. MR 1308945 |
| Reference:
|
[13] C. Boyd: Preduals of the space of holomorphic functions on a Fréchet space.Ph.D. Thesis, National University of Ireland (1992). |
| Reference:
|
[14] C. Boyd: Distinguished preduals of the space of holomorphic functions.Rev. Mat. Univ. Complut. Madrid 6 (1993), 221–232. MR 1269753 |
| Reference:
|
[15] C. Boyd: Montel and reflexive preduals of the space of holomorphic functions.Studia Math. 107 (1993), 305–325. MR 1247205, 10.4064/sm-107-3-305-315 |
| Reference:
|
[16] C. Boyd: Some topological properties of preduals of spaces of holomorphic functions.Proc. Royal Ir. Acad. 94A (1994), 167–178. Zbl 0852.46037, MR 1369030 |
| Reference:
|
[17] C. Boyd and A. Peris: A projective description of the Nachbin ported topology.J. Math. Anal. Appl. 197 (1996), 635–657. MR 1373071, 10.1006/jmaa.1996.0044 |
| Reference:
|
[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 0295022 |
| Reference:
|
[19] H. Collins and W. Ruess: Duals of spaces of compact operators.Studia Math. 74 (1982), 213–245. MR 0683747, 10.4064/sm-74-3-213-245 |
| Reference:
|
[20] A. Defant and W. Govaerts: Tensor products and spaces of vector-valued continuous functions.Manuscripta Math. 55 (1986), 432–449. MR 0836875 |
| Reference:
|
[21] S. Dineen: Complex Analysis on Locally Convex Spaces.North-Holland Math. Studies, Vol. 57, 1981. MR 0640093 |
| Reference:
|
[22] S. Dineen: Holomorphic functions and Banach-nuclear decompositions of Fréchet spaces.Studia Math. 113 (1995), 43–54. Zbl 0831.46048, MR 1315520, 10.4064/sm-113-1-43-54 |
| Reference:
|
[23] K. G. Große-Erdmann: The Borel-Okada theorem revisited.Habilitation Schrift, 1992. |
| Reference:
|
[24] R. B. Holmes: Geometric Functional Analysis and its Applications.Springer-Verlag Graduate Texts in Math., Vol. 24, 1975. Zbl 0336.46001, MR 0410335, 10.1007/978-1-4684-9369-6 |
| Reference:
|
[25] J. Horváth: Topological Vector Spaces and Distributions, Vol. 1.Addison-Wesley, Massachusetts, 1966. MR 0205028 |
| Reference:
|
[26] H. Jarchow: Locally Convex Spaces.B. G. Teubner, Stuttgart, 1981. Zbl 0466.46001, MR 0632257 |
| Reference:
|
[27] N. Kalton: Schauder decompositions in locally convex spaces.Cambridge Phil. Soc. 68 (1970), 377–392. Zbl 0196.13505, MR 0262796, 10.1017/S0305004100046193 |
| Reference:
|
[28] J. Mujica: A Banach-Dieudonné Theorem for the space of germs of holomorphic functions.J. Funct. Anal. 57 (1984), 32–48. MR 0744918, 10.1016/0022-1236(84)90099-5 |
| Reference:
|
[29] J. Mujica and L. Nachbin: Linearization of holomorphic mappings on locally convex spaces.J. Math. Pures Appl. 71 (1992), 543–560. MR 1193608 |
| Reference:
|
[30] L. Nachbin: Weak holomorphy.Unpublished manuscript. |
| Reference:
|
[31] P. Pérez Carreras and J. Bonet: Barrelled Locally Convex Spaces. North Holland Math. Studies, Vol. 131.North Holland, Amsterdam, 1987. MR 0880207 |
| Reference:
|
[32] A. Peris: Quasinormable spaces and the problem of topologies of Grothendieck.Ann. Acad. Sci. Fenn. 19 (1994), 167–203. Zbl 0789.46006, MR 1246894 |
| Reference:
|
[33] A. Peris: Topological tensor products of a Fréchet Schwartz space and a Banach space.Studia Math. 106 (1993), 189–196. Zbl 0810.46008, MR 1240313, 10.4064/sm-106-2-189-196 |
| Reference:
|
[34] W. Ruess: Compactness and collective compactness in spaces of compact operators.J. Math. Anal. Appl. 84 (1981), 400–417. Zbl 0477.47027, MR 0639673, 10.1016/0022-247X(81)90177-3 |
| Reference:
|
[35] R. A. Ryan: Applications of topological tensor products to infinite dimensional holomorphy.Ph.D. Thesis, Trinity College, Dublin, 1980. |
| Reference:
|
[36] H. H. Schaefer: Topological Vector Spaces.Third Printing corrected, Springer-Verlag, 1971. Zbl 0217.16002, MR 0342978 |
| Reference:
|
[37] J. Schmets: Espaces de fonctions continues.Lecture Notes in Math., Vol. 519, Springer-Verlag, Berlin-Heidelberg-New York, 1976. Zbl 0334.46022, MR 0423058 |
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