In mathematics, with special application to complex analysis, a normal family is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. Note that a compact family of continuous functions is automatically a normal family. Sometimes, if each function in a normal family F satisfies a particular property (e.g. is holomorphic), then the property also holds for each limit point of the set F.

More formally, let X and Y be topological spaces. The set of continuous functions ${\displaystyle f:X\to Y}$ has a natural topology called the compact-open topology. A normal family is a pre-compact subset with respect to this topology.

If Y is a metric space, then the compact-open topology is equivalent to the topology of compact convergence, and we obtain a definition which is closer to the classical one: A collection F of continuous functions is called a normal family if every sequence of functions in F contains a subsequence which converges uniformly on compact subsets of X to a continuous function from X to Y. That is, for every sequence of functions in F, there is a subsequence ${\displaystyle f_{n}(x)}$ and a continuous function ${\displaystyle f(x)}$ from X to Y such that the following holds for every compact subset K contained in X:

where ${\displaystyle d_{Y}}$ is the metric of Y.

The concept arose in complex analysis, that is the study of holomorphic functions. In this case, X is an open subset of the complex plane, Y is the complex plane, and the metric on Y is given by ${\displaystyle d_{Y}(y_{1},y_{2})=|y_{1}-y_{2}|}$. As a consequence of Cauchy's integral theorem, a sequence of holomorphic functions that converges uniformly on compact sets must converge to a holomorphic function. That is, each limit point of a normal family is holomorphic.

Normal families of holomorphic functions provide the quickest way of proving the Riemann mapping theorem.

More generally, if the spaces X and Y are Riemann surfaces, and Y is equipped with the metric coming from the uniformization theorem, then each limit point of a normal family of holomorphic functions ${\displaystyle f:X\to Y}$ is also holomorphic.

For example, if Y is the Riemann sphere, then the metric of uniformization is the spherical distance. In this case, a holomorphic function from X to Y is called a meromorphic function, and so each limit point of a normal family of meromorphic functions is a meromorphic function.