For questions about normal families and their properties. A normal family is a precompact family of continuous functions.
In complex analysis, a collection $\Phi$ of holomorphic functions defined on some open set $U$ is called a normal family if every sequence of functions in $\Phi$ has a subsequence which converges compactly. That is, there is a subsequence which converges uniformly on every compact set $K \subseteq U$.
More generally, a family $F$ of continuous functions from a complete metric space $X$ to another complete metric space $Y$ is normal if every sequence in $F$ has a subsequence which converges uniformly on compact subsets of $X$ to a continuous function from $X$ to $Y$.
Reference: Normal family.