Interdependence of chapters -- Courses and Seminars: Guide to the Essentials of Specific Topics -- Part One: Function Theory on Domains in ℂn -- Chapter 0. Elementary Properties of Holomorphic Functions -- 1 Definition of Holomorphic Function -- 2 O (X) as a Topological Algebra -- 3 Holomorphic Mappings -- 4 Cauchy's Integral Formula -- 4A Supplement: Cartan's Uniqueness Theorem and Automorphisms of Bounded Domains -- 5 Montel's Theorem -- 6 The Identity Theorem, the Maximum Principle, and Runge's Theorem -- 7 The Riemann Removable Singularity Theorems -- 8 The Implicite Mapping Theorem -- Chapter 1 Regions of Holomorphy -- 11 Domains of Convergence of Power Series and Reinhardt Domains -- 12 Regions of Holomorphy and Holomorphic Convexit -- 12A Supplement: Further Extension Theorems for Holomorphic Functions -- 12B Supplement: The Edge-of-the-Wedge Theorem -- 13 Plurisubharmonic Functions -- 14 Pseudoconvex Domains -- Part Two: Function Theory on Analytic Sets -- Chapter 2 The Weierstrass Preparation Theorem and its Applications -- 21 Power Series Algebras -- 22 The Weierstrass Theorems -- 23 Algebraic Properties of nO0 and -- 23A Supplement: Finite Ring-extensions, Normality -- 24 Analytic Algebras -- Chapter 3 Complex Manifolds and the Elementary Theory of Complex Spaces -- 30 Bringing in the Sheaves -- 31 Ringed Spaces and Local Models of Complex Spaces -- 32 Complex Manifolds and Reduced Complex Spaces -- 32A Supplement: Submersions and Immersions -- 32B Supplement: Examples of Complex Manifolds -- 33 Zeros of Polynomials -- 33A Supplement: Resultants and Discriminants -- 33B Supplement: Proper Mappings and Equivalence Relations -- Chapter 4 Complex Spaces -- 41 Coherent Sheaves -- 42 The Coherence of nO -- 43 Complex Spaces -- 44 Germs of Complex Spaces and Analytic Algebras.

45 Discrete and Finite Holomorphic Mappings -- 45A Supplement: Image Sheaves -- 45B Supplement: The Analytic Spectrum -- 46 The Representation Theorem for Prime Germs -- 46A Supplement: Injective Holomorphic Mappings between Manifolds of the Same Dimension -- 46B Supplement: Universal Denominators for Prime Germs -- 47 Hilbert's Nullstellensatz and Cartan's Coherence Theorem -- 48 Dimension Theory -- 49 Set of Singular Points and Decomposition into Irreducible Components -- 49A Supplement: Fiber Products and Quotients -- Part Three: Function Theory on Stein Spaces -- Chapter 5 Applications of Theorem B -- 50 Introductory Remarks on Cohomology -- 50A Supplement: Automorphic Functions -- 51 Stein Spaces -- 51A Supplement: Countable Topology in Complex Spaces -- 52 Theorem B and B-Spaces -- 53 The Additive Cousin Problem -- 53A Supplement: Meromorphic Functions on Reduced Complex Spaces -- 54 The Multiplicative Cousin Problem -- 54 A Supplement: The Poincaré Problem -- 54B Supplement: Holomorphic Line Bundles -- 55 Coherent Analytic Sheaves as Fréchet Sheaves -- 56 The Exhaustion Theorem -- 57 The Character Theorem and Holomorphic Hulls -- 58 The Holomorphic Version of de Rham's Theorem -- 58A Supplement: The Grassmann Algebra and Differential Forms -- Chapter 6 Proof of Theorem B -- 61 Dolbeault's Lemma -- 62 Theorem B for Strictly Pseudoconvex Domains -- 63 Characterization of Stein Spaces -- 63A Supplement: Levi's Problem for Pseudoconvex Domains -- 63B Supplement: Weakly Holomorphically Convex Spaces are Holomorphically Convex -- Supplement: Chapter 7 Normal Complex Spaces -- 71 Normalization -- 72 Maximal Complex Structure -- 73 Finite Mappings on Stein Spaces -- 74 A Criterium for Normality -- List of Examples -- Bibliography -- Glossary of Notations -- Index.