×

Holomorphic mappings on \(\ell _ 1\). (English) Zbl 0637.46045

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

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
PDFBibTeX XMLCite
Full Text: DOI