Behnke-Stein theorem

Theorem (Behnke-Stein).

Suppose that $G_{k}\subset{\mathbb{C}}^{n}$ are domains of holomorphy such that $G_{k}\subset G_{j}$ whenever $k<j$ . Then $\bigcup_{k=1}^{\infty}G_{k}$ is a domain of holomorphy.

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.

References

1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title	Behnke-Stein theorem
Canonical name	BehnkeSteinTheorem
Date of creation	2013-03-22 14:31:13
Last modified on	2013-03-22 14:31:13
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Theorem
Classification	msc 32T05
Synonym	increasing union of domains of holomorphy is a domain of holomorphy