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. 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. 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. 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 f : X → Y {displaystyle f:X o Y} has a natural topologycalled 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 of that 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 f n ( x ) {displaystyle f_{n}(x)} and a continuous function f ( x ) {displaystyle f(x)} from X to Y such that the following holds for every compact subset K contained in X: lim n → ∞ sup x ∈ K d Y ( f n ( x ) , f ( x ) ) = 0 {displaystyle lim _{n ightarrow infty }sup _{xin K}d_{Y}(f_{n}(x),f(x))=0} where d Y ( y 1 , y 2 ) {displaystyle d_{Y}(y_{1},y_{2})} is the distance metric associated with the metric space Y. The concept arose in complex analysis, that is the study 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 d Y ( y 1 , y 2 ) = | y 1 − y 2 | {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. 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 f : X → Y {displaystyle f:X o 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 pointof a normal family of meromorphic functions is a meromorphic function.