Real analytic curves in Frechet spaces and their duals

The following results are presented: 1) a characterization through the Liouville property of those Stein manifolds U such that every germ of holomorphic functions on R x U can be developed locally as a vector-valued Taylor series in the first variable with values in H(U); 2) if T(mu) is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solution u of the equation T(mu)u = f which depends holomorphically on the parameter z is an element of C whenever f 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-spaces E for which E-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.