# Definition:Normal Family

## Definition

Let $X = \struct {M_1, d_1}$ and $Y = \struct {M_2, d_2}$ be complete metric spaces.

Let $\FF = \family {f_i}_{i \mathop \in I}$ be a family of continuous mappings $f_i: X \to Y$.

Then $\FF$ is a normal family if and only if:

every sequence of mappings in $\FF$ contains a subsequence which converges uniformly on compact subsets of $X$ to a continuous function $f: X \to Y$.