# 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 is defined as follows.

Let be a domain (an open and connected set), or alternatively for a more general definition, let be an dimensional complex analytic manifold. Further let stand for the set of holomorphic functions on For a compact set , the **holomorphically convex hull** of is

One obtains a narrower concept of **polynomially convex hull** by taking instead to be the set of complex-valued polynomial functions on *G*. The polynomially convex hull contains the holomorphically convex hull.

The domain is called **holomorphically convex** if for every compact subset is also compact in . Sometimes this is just abbreviated as *holomorph-convex*.

When , any domain is holomorphically convex since then is the union of with the relatively compact components of . Also, being holomorphically convex is the same as being a domain of holomorphy (The Cartan–Thullen theorem). These concepts are more important in the case of several complex variables (*n* > 1).

## See also

## 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.

*This article incorporates material from Holomorphically convex on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*