Holomorphic functions with values in locally convex spaces and applications to integral formulas

LUTZ BUNGART1. Introduction.Holomorphic functions of one variable with values in a locally convex (Hausdorff topological vector) space (over the field C of complex numbers) have been studied before by A. Grothendieck [11].However, A. Grothendieck is mainly concerned with the characterisation of the (topological) dual space of the space of holomorphic functions on an open (resp.compact) subset of the Riemann sphere having values in a locally convex space; and so are subsequent authors who generalize his ideas to Riemann surfaces and multicircular domains in several complex variables.The purpose of our exposition is entirely different.We are concerned with properties of holomorphic functions and spaces of holomorphic functions with values in locally convex spaces which are defined on analytic spaces.We want to extend the basic results of H. Cartan [5] to this more general class of functions and give some applications.Some of these problems have already been considered and solved in the existing literature for the very special case of families of holomorphic functions depending continuously on a parameter t e [0,1].Such families of holomorphic functions can be considered as holomorphic functions with values in the Banach space ^([0,1]) of continuous functions on the unit interval [0,1].We have to use some results from the theory of locally convex spaces not all of which are easily accessible.We recall some of the proofs (Proposition 3.1 and Lemma 5.1) for convenience.In the first part of this thesis Theorems A and B of H. Cartan are generalized to analytic sheaves which are coherent over the sheaf of germs of holomorphic functions (on a Stein space (X, &)) with values in some Frechet space £.These coherent analytic sheaves are of the form £f e E, where S? is a coherent analytic sheaf on X and if z E is the sheaf defined by the presheaf {H(U, Sf) e £, U cz X open}.The s-product used in the definition is the one defined by L. Schwartz; it is similar to the tensor product in algebra.For £, F locally convex spaces, F ® £ is contained in F sE.We usually assume that one of the spaces, say P, has a certain (nuclearity) property (~V).This is the case for instance if F is taken