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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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Keywords: Holomorphic mapping, monomial expansion, lifting, Nachbin topology
Article copyright: © Copyright 1987 American Mathematical Society