Ryan, Raymond A. Holomorphic mappings on \(\ell _ 1\). (English) Zbl 0637.46045 Trans. Am. Math. Soc. 302, 797-811 (1987). The purpose of this article is to give a complete description of the holomorphic mappings from the Banach space \(\ell_ 1\) over the complex field into an arbitrary Banach space over the same field. 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 natural topologies on the space \({\mathcal H}(\ell_ 1;X)\) of holomorphic mappings, and to prove a lifting property for holomorphic mappings on \(\ell_ 1\). It is also shown that the monomials form an unconditional basis in the Schauder sense for the space \({\mathcal H}(\ell_ 1)\) of holomorphic functions with the compact open topology. Reviewer: L.Nachbin Cited in 1 ReviewCited in 24 Documents MSC: 46G20 Infinite-dimensional holomorphy 46E10 Topological linear spaces of continuous, differentiable or analytic functions 32A30 Other generalizations of function theory of one complex variable 58B12 Questions of holomorphy and infinite-dimensional manifolds Keywords:holomorphic mappings from the Banach space \(\ell _ 1\) over the complex field into an arbitrary Banach space over the same field; monomial expansions; growth of the norms of the coefficients; lifting property for holomorphic mappings; unconditional basis; compact open topology PDFBibTeX XMLCite \textit{R. A. Ryan}, Trans. Am. Math. Soc. 302, 797--811 (1987; Zbl 0637.46045) Full Text: DOI