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Real analytic curves in Fréchet spaces and their duals. (English) Zbl 0918.46034

Summary: 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 \(\mathbb{R}\times U\) 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 analysis functions, one can find a solution \(u\) of the equation \(T_\mu u=f\) which depends holomorphically on the parameter \(z\in\mathbb{C}\) 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 Fréchet 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.

MSC:

46E40 Spaces of vector- and operator-valued functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46M05 Tensor products in functional analysis
46M15 Categories, functors in functional analysis
46N20 Applications of functional analysis to differential and integral equations
35R50 PDEs of infinite order
32A10 Holomorphic functions of several complex variables
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A32 Spaces of linear operators; topological tensor products; approximation properties
32A05 Power series, series of functions of several complex variables
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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