Behnke–Stein theorem

{{Short description|Theorem in mathematics about unions of domains of holomorphy}}

{{confuse|Behnke–Stein theorem on Stein manifolds}}

In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence $G\_k\; \backslash subset\; \backslash mathbb\{C\}^n$ (i.e., $G\_k\; \backslash subset\; G\_\{k\; +\; 1\}$) of domains of holomorphy is again a domain of holomorphy. It was proved by Heinrich Behnke and Karl Stein in 1938.{{cite journal |last1=Behnke |first1=H. |authorlink1=Heinrich Behnke |last2=Stein |first2=K. |authorlink2=Karl Stein (mathematician) |title=Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität |journal=Mathematische Annalen |date=1939 |volume=116 |pages=204–216 |doi=10.1007/BF01597355|s2cid=123982856 }}

This is related to the fact that an increasing union of pseudoconvex domains is pseudoconvex and so it can be proven using that fact and the solution of the Levi problem. Though historically this theorem was in fact used to solve the Levi problem, and the theorem itself was proved using the Oka–Weil theorem. This theorem again holds for Stein manifolds, but it is not known if it holds for Stein space.{{cite arXiv |eprint=0905.2343|last1=Coltoiu|first1=Mihnea|title=The Levi problem on Stein spaces with singularities. A survey|year=2009|class=math.CV}}