Analytic functions with values in a Frechet space

Abstract

where {Pn} is a sequence of continuous mutually annihilating projections on F whose ranges are all one-dimensional subspaces of F. This repre­ sentation reduces the study of , which are essentially scalar-valued analytic functions. We actually prove the stronger (and more useful) result that if { k. Expansions of vector-valued functions of a different type have been considered by Grothendick [6]. Theorem 1 is applied to generalize Theorem B of H. Cartan [3]. We consider a coherent analytic sheaf S on a Stein manifold M and introduce the notion of the vectorization SF of S (relative to a given Frechet space F). If 0 denotes the sheaf of locally-defined analytic functions and 0F denotes the sheaf of locally-defined analytic functions with values in Ff then SF is defined to be the tensor product S 0 0F of the 0-modules S and 0F. For the important case of a coherent analytic subsheaf S of the sheaf 0 of locally-defined fc-tuples of analytic functions, SF turns out to be canonically isomorphic to the sheaf SF determined by assigning to each open set U the module of all fc-tuples (fu • • •, fk) of analytic functions from U to F which have the property that for each u in F * the fc-tuple (uofu . . . , uofk) is a cross-section of S over U. For instance, if S is the sheaf of all locally-defined analytic functions which vanish on a given analytic set A then it is evident that S'F is the sheaf of all locally-defined analytic functions with values in F which vanish on A. One of the main results, an extension of Theorem B of [3], will be that the cohomology groups H(M, SF) vanish in all dimensions N ^ 1, where SF is the vectorization of a coherent analytic sheaf S on a Stein manifold M. Using this theorem and the isomorphism of SF to the sheaf SF defined above one could show, for instance, that the usual