Real analytic curves in Fréchet spaces and their duals

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 Frechet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.