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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Holomorphic mappings on $l_ 1$
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by Raymond A. Ryan PDF
Trans. Amer. Math. Soc. 302 (1987), 797-811 Request permission


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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 302 (1987), 797-811
  • MSC: Primary 46G20; Secondary 32A15, 58B12
  • DOI:
  • MathSciNet review: 891648