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Real analytic curves in Fréchet spaces and their duals

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Abstract

The following results are presented: 1) a characterization through the Liouville property of those Stein manifoldsU such that every germ of holomorphic functions on ℝ xU can be developed locally as a vector-valued Taylor series in the first variable with values inH(U); 2) ifT μ is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solutionu of the equationT μ u=f which depends holomorphically on the parameterz∈ℂ wheneverf depends in the same manner. These results are obtained as an application of a thorough study of vector-valued real analytic maps by means of the modern functional analytic tools. In particular, we give a tensor product representation and a characterization of those Fréchet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, E.T.S. Arquitectura, E-46071, Valencia, Spain

    José Bonet

  2. Faculty of Mathematics and Comp. Sci., A. Mickiewicz University Poznań, Matejki 48/49, 60-769, Poznań, Poland

    Pawel Domański

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  1. José Bonet
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  2. Pawel Domański
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Bonet, J., Domański, P. Real analytic curves in Fréchet spaces and their duals. Monatshefte für Mathematik 126, 13–36 (1998). https://doi.org/10.1007/BF01312453

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