In mathematics, especially
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...
, the Behnke–Stein theorem states that a union of an increasing sequence
(i.e.,
) of
domains of holomorphy is again a domain of holomorphy. It was proved by
Heinrich Behnke and
Karl Stein in 1938.
This is related to the fact that an increasing union of
pseudoconvex domain
In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the ''n''-dimensional complex space C''n''. Pseudoconvex sets are important, as they allow for classificatio ...
s 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
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil.
Statement
The Oka–Weil theorem sta ...
. This theorem again holds for Stein manifolds, but it is not known if it holds for Stein space.
References
*
*
Several complex variables
Theorems in complex analysis
{{mathanalysis-stub