Wikipedia gives Montel's theorem saying essentially $\mathcal F$ is a normal family in $\mathbb H(G)$ iff $\mathcal F$ is uniformly bounded. My book say the same thing but iff $\mathcal F$ is locally (uniformly) bounded. Wikipedia lets $G$ be arbitrary open, whereas my book says that $G$ is a domain (connected). Presumably they are both correct. Does this mean that we can show that if $\mathcal F$ is locally bounded on a domain, then it is uniformly bounded? What's going on?
The version of Montel's theorem I can find on Wikipedia says:
The first, and simpler, version of the theorem states that a uniformly bounded family of holomorphic functions defined on an open subset of the complex numbers is normal.
I.e. just an implication, not an equivalence. This implication is true. Clicking onward to the page on normal families, we find
Montel's theorem asserts that every locally bounded family of holomorphic functions is normal.
This is also correct. In other words locally (uniform) boundedness is enough. None of the Wikipedia pages I find say anything about the converse.
(And of course, locally uniform boundedness does not imply uniform boundedness: Take for example $f_n(z) = nz^n$ on the unit disc.)