Projective limits of inductive limits of Banach spaces, so-called (PLB)-spaces, arise naturally in analysis. For instance the space of distributions, the space of real analytic functions and several spaces of ultradifferentiable functions and ultradistributions are of this type. In this thesis we study (PLB)-spaces whose building blocks are Banach spaces of holomorphic functions defined by a weighted sup-norm. The investigation of these spaces extends research of Agethen, Bierstedt, Bonet who recently studied weighted (PLB)-spaces of continuous functions. From another perspective, it extends the study of weighted inductive limits of Banach spaces of holomorphic functions, which have been studied intensely during the last years by several authors.
Our aim concerning the spaces described above is to study locally convex properties like ultrabornologicity or barrelledness. As the starting point in the definition of the spaces under investigation is a double sequence of strictly positive and continuous functions (weights), our aim is to characterize the forementioned properties in terms of this sequence. In addition, we investigate under which circumstances projective and inductive limit can be interchanged and therefore the (PLB)-space coincides with the weighted inductive limit of Fréchet spaces defined by the same sequence; spaces of the latter type have been investigated by Bierstedt, Bonet.
We prove necessary conditions for the forementioned properties of the spaces and for the interchangeability of projective and inductive limit under rather mild assumptions. Concerning sufficient conditions we make use of homological methods, whose exploration was started by Palamodov in the late sixties and carried on by Vogt, Wengenroth and many others during the last 40 years. For technical reasons the methods just mentioned do not apply to all cases which we want to study. Thus, we first present a criterion for barrelledness adjusted to these situations. However, it seems to be inevitable to decompose holomorphic functions to prove any result concerning sufficient conditions at all. Therefore we introduce several settings in which the latter is possible; within these settings the decomposition is achieved in different ways, namely by the decomposition of polynomials (on the disc and on the plane), a method connected with the theory of Bergman projections, two types of sequence space representations and Hörmander's dbar-method. Under some additional assumptions (which are - as we show - satisfied in many examples) we finally provide in almost all settings mentioned above a full characterization of ultrabornologicity, barrelledness and interchangeability of projective and inductive limit.
To accomplish our investigation of weighted (PLB)-spaces, we present two results (one for continuous and one for holomorphic functions) which show that spaces of this type can sometimes be written as a tensor product of a Fréchet space with a (DF)-space. Combined with results on weighted (PLB)-spaces the result on continuous functions is connected to work of Grothendieck, who studied ultrabornologicity of such kinds of tensor products. The second result on tensor product representations exhibits that some of the so-called mixed spaces of ultradistributions (introduced recently by Schmets and Valdivia) happen to be weighted (PLB)-spaces of holomorphic functions.