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.
If Y is a metric space, then the compact-open topology is equivalent to the topology of compact convergence,[1] 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.
The concept arose in complex analysis, that is the study of holomorphic functions.
Normal families of holomorphic functions provide the quickest way of proving the Riemann mapping theorem.
[2] 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
In the classical context of holomorphic functions, there are several criteria that can be used to establish that a family is normal: Montel's theorem states that a family of locally bounded holomorphic functions is normal.
The Montel-Caratheodory theorem states that the family of meromorphic functions that omit three distinct values in the extended complex plane is normal.
Marty's theorem[3] provides a criterion equivalent to normality in the context of meromorphic functions: A family
to the complex plane is a normal family if and only if for each compact subset K of U there exists a constant C so that for each
and each z in K we have Indeed, the expression on the left is the formula for the pull-back of the arclength element on the Riemann sphere to the complex plane via the inverse of stereographic projection.
[4][5] Because the concept of a normal family has continually been very important to complex analysis, Montel's terminology is still used to this day, even though from a modern perspective, the phrase pre-compact subset might be preferred by some mathematicians.
Note that though the notion of compact open topology generalizes and clarifies the concept, in many applications the original definition is more practical.