Let \(\Omega\subseteq\CC\) be open. A family \(\mathcal F\) of holomorphic functions on \(\Omega\) is normal if every sequence in \(\mathcal F\) has a subsequence converging uniformly on every compact subset of \(\Omega\).
The limit of such a subsequence need not lie in \(\mathcal F\): normality says that \(\mathcal F\) is precompact in the topology of locally uniform convergence, and \(\mathcal F\) is compact in that topology exactly when, in addition, every such limit lies in \(\mathcal F\).
In the proof of the Riemann mapping theorem for a simply connected open set \(\Omega\subsetneq\CC\) and \(z_0\in\Omega\), one maximizes \(\abs{f'(z_0)}\) over the injective holomorphic maps \(f\colon\Omega\to\DD\) with \(f(z_0)=0\). This family is uniformly bounded, hence normal by Montel's theorem, so a maximizing sequence has a locally uniformly convergent subsequence, whose limit is the candidate Riemann map.
Two inequivalent definitions of normality are in use. The definition by locally uniformly convergent subsequences is that of [@SS03]. In [@Ahl79], a family is normal if every sequence has a subsequence that either converges uniformly on compact sets or tends uniformly to \(\infty\) on compact sets; this is locally uniform convergence in the spherical metric on the Riemann sphere, and the same definition applies to families of meromorphic functions.
The family \(\{f_n\}_{n\ge1}\) with \(f_n(z)=n(z^2-n)\) on \(\CC\) is normal in the sense of [@Ahl79], and the sequence \((f_n)\) has no locally uniformly convergent subsequence. On \(\abs{z}\le R\) and for \(n>R^2\), \(\abs{f_n(z)}\ge n^2-nR^2\to\infty\) uniformly, so every subsequence tends uniformly to \(\infty\) on compact sets and none converges uniformly on compact sets to a finite function.
Montel's theorem is [@SS03, Theorem 3.3].