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, 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. These concepts are more important in the case of several complex variables.
