I am looking for a proper way to define distane on the space of Holomorphic functions defined on a domain $D$.Does the Montel's Theorem (Given below from Stein's Book) helps to Characterize Compact Sets in the space of Holomorphic function with this distace ?
Montel's Theorem States That:
Theorem: Suppose $F$ be a family of holomorphic functions defined on a domain $D$ which is uniformly bounded on every compact subset of $D$.Then:
1.The family $F$ is equicontinuous on every compact subset of $D$
2.The Family $F$ is normal.
Please help.Thanks in Advance!
The space of analytic functions is a subspace of the space of continuous functions. The metric on this space is defined as follows.
If $G$ is an open set in $\mathbb C$, exhaust $G$ by a sequence of compact sets. Define $\rho_n = \sup \{ d(f(z), g(z)):z\in K_n\}$. Then we can define the metric as $$\rho (f,g) = \sum_{n=1}^{\infty} (\frac{1}{2})^n \frac {\rho_n (f,g)}{1+\rho_n (f,g)}$$.
Source: Conway's Functions of One Complex Variable.