Normal family

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,^[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. 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:

${\displaystyle \lim _{n\rightarrow \infty }\sup _{x\in K}d_{Y}(f_{n}(x),f(x))=0}$

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