Holomorphically convex hull

In mathematics, more precisely in complex analysis, the holomorphically convex hull of a given compact set in the "n"-dimensional complex space C^"n" is defined as follows.

Let $G\; subset\; \{mathbb\{C^n$ be a domain (an open and connected set), or alternatively for a more general definition, let $G$ be an $n$ dimensional complex analytic manifold. Further let $\{mathcal\{O(G)$ stand for the set of holomorphic functions on $G.$ For a compact set $K\; subset\; G$, the holomorphically convex hull of $K$ is

:

(One obtains a narrower concept of polynomially convex hull by requiring in the above definition that "f" be a polynomial.)

The domain $G$ is called holomorphically convex if for every $K\; subset\; G$ compact in $G$, $hat\{K\}\_G$ is also compact in $G$. Sometimes this is just abbreviated as "holomorph-convex".

When $n=1$, any domain $G$ is holomorphically convex since then $hat\{K\}\_G$ is the union of $K$ with the relatively compact components of $G\; setminus\; K\; subset\; G$. Also note that for "n"=1 being holomorphically convex is the same as being a domain of holomorphy. These concepts are more important in the case "n" > 1 of several complex variables.

ee also

* Stein manifold
* Pseudoconvexity

References

* Lars Hörmander. "An Introduction to Complex Analysis in Several Variables", North-Holland Publishing Company, New York, New York, 1973.
* Steven G. Krantz. "Function Theory of Several Complex Variables", AMS Chelsea Publishing, Providence, Rhode Island, 1992.----