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Transactions of the American Mathematical Society

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Holomorphic mappings on $ l\sb 1$


Author: Raymond A. Ryan
Journal: Trans. Amer. Math. Soc. 302 (1987), 797-811
MSC: Primary 46G20; Secondary 32A15, 58B12
MathSciNet review: 891648
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Abstract: We describe the holomorphic mappings of bounded type, and the arbitrary holomorphic mappings from the complex Banach space $ {l_1}$ into a complex Banach space $ X$. It is shown that these mappings have monomial expansions and the growth of the norms of the coefficients is characterized in each case. This characterization is used to give new descriptions of the compact open topology and the Nachbin ported topology on the space $ \mathcal{H}({l_1};X)$ of holomorphic mappings, and to prove a lifting property for holomorphic mappings on $ {l_1}$. We also show that the monomials form an equicontinuous unconditional Schauder basis for the space $ (\mathcal{H}({l_1}),{\tau _0})$ of holomorphic functions on $ {l_1}$ with the topology of uniform convergence on compact sets.


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  • [1] Richard M. Aron, Luiza A. Moraes, and Raymond A. Ryan, Factorization of holomorphic mappings in infinite dimensions, Math. Ann. 277 (1987), no. 4, 617–628. MR 901708, 10.1007/BF01457861
  • [2] Philip J. Boland and Séan Dineen, Holomorphic functions on fully nuclear spaces, Bull. Soc. Math. France 106 (1978), no. 3, 311–336 (English, with French summary). MR 515406
  • [3] Philip J. Boland and Seán Dineen, Duality theory for spaces of germs and holomorphic functions on nuclear spaces, Advances in holomorphy (Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) North-Holland, Amsterdam, 1979, pp. 179–207. North-Holland Math. Studies, 34. MR 0632038
  • [4] Soo Bong Chae, Holomorphy and calculus in normed spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 92, Marcel Dekker, Inc., New York, 1985. With an appendix by Angus E. Taylor. MR 788158
  • [5] S. Dineen, Holomorphic functions on nuclear spaces, Proc. 2nd Paderborn Conference on Functional Analysis (K. D. Bierstedt and B. Fuchssteiner, Eds.), Math. Studies, no. 38, North-Holland, Amsterdam, 1979, pp. 317-326.
  • [6] Seán Dineen, Analytic functionals on fully nuclear spaces, Studia Math. 73 (1982), no. 1, 11–32. MR 673336
  • [7] Seán Dineen, Complex analysis in locally convex spaces, North-Holland Mathematics Studies, vol. 57, North-Holland Publishing Co., Amsterdam-New York, 1981. Notas de Matemática [Mathematical Notes], 83. MR 640093
  • [8] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0891648-7
Keywords: Holomorphic mapping, monomial expansion, lifting, Nachbin topology
Article copyright: © Copyright 1987 American Mathematical Society