The original motivation for writing this paper was to give a proof for the existence of the moduli space of simple holomorphic bundles 6 (i.e. End(£)=C) on a fixed compact complex space X. This is achieved in (6.5). The method, used by us, is so general that we can formulate an abstract sufficient criterion for the representability of analytic functors (see theorem (2.2)). Even simple coherent sheaves can be treated in this way (theorem (6.4)). Some special cases were obtained formerly by A. Norton in [No], where X has to be a manifold. A more explicit construction of the moduli space of simple bundles in the kahlerian case is given in [Lii-Ok] by a completely different approach.Our representability theorem (2.2) is in some sense more special than the one's known in the literature (for example see [Bi] 2 or [S-V]), but has the advantage to fit into the philosophy of constructing moduli spaces as quotients with respect to group operations. It is also necessary here to consider relative group operations on complex spaces and to construct relative quotients. The material that we need, is contained in § 3.We also deal with the existence of coarse moduli spaces of analytic functors in general. This notion is weaker than that of the representability of the associated sheafified functor. The main result is formulated in theorem (2.3). It should be mentioned that this theorem is inspired by the work of G. Schumacher, see [Schu]This paper grew out from discussions of both authors at the Sonderforschungsbereich 170 "Geometric und Analysis" in Gottingen. We would like to express our sincere thanks to the SFB, especially for the stimulating atmosphere. Last not least, H. Flenner was very interested in our work and we want to thank him for several useful remarks.

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Using convex subrings of *$\mathbb{C}$, a nonstandard extension of $\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

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In this note we identify two complex structures (one is given by algebraic geometry, the other by gauge theory) on the set of isomorphism classes of holomorphic bundles with section on a given compact complex manifold. In the case of line bundles, these complex spaces are shown to be isomorphic to a space of effective divisors on the manifold. * Partially supported by SNF, nr. 2000-055290.98/1