Definition:Normal Family

Definition
Let $X = \left({M_1, d_1}\right)$ and $Y = \left({M_2, d_2}\right)$ be complete metric spaces.

Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of continuous functions $f_i: X \to Y$.

Then $\mathcal F$ is called a normal family if every sequence of functions in $\mathcal F$ contains a subsequence which converges uniformly on compact subsets of $\mathcal F$ to a continuous function $f_i: X \to Y$.