Normal family

In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions with respect to the compact-open topology. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. It is of general interest to understand compact sets in function spaces, since these are usually truly infinite-dimensional in nature.

More formally, a family (equivalently, a set) F of continuous functions f defined on some complete metric space X with values in another complete metric space Y is called normal 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:

...
Wikipedia