holomorphically convex

Let $G\subset {ℂ}^n$ be a domain, or alternatively for a more general definition let $G$ be an $n$ dimensional complex analytic manifold.Further let $𝒪(G)$ stand for the set of holomorphic functions on $G$.

Definition.

Let $K\subset G$ be a compact set.We define the holomorphically of $K$ as

$${\widehat{K}}_G:=\{z\in G\mid |f(z)|\le \underset{w\in K}{sup}|f(w)|\text{ for all }f\in 𝒪(G)\}.$$

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

Note that when $n=1$, any domain $G$ is holomorphically convex since when $n=1$ ${\widehat{K}}_G=K$ for all compact $K\subset G$. Also note that this is the same as being a domain of holomorphy.