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 f : X → Y {\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 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\rightarrow \infty }\sup _{x\in K}d_{Y}(f_{n}(x),f(x))=0}
where d Y {\displaystyle d_{Y}} is the metric of Y.
Normal families of holomorphic functions 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 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. 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 f : X → Y {\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.
Criteria 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. For a family of holomorphic functions, this reduces to requiring two values omitted by viewing each function as a meromorphic function omitting the value infinity. Marty's theorem provides a criterion equivalent to normality in the context of meromorphic functions: A family F {\displaystyle F} of meromorphic functions from a domain U ⊂ C {\displaystyle U\subset \mathbb {C} } 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 f ∈ F {\displaystyle f\in F} and each z in K we have
2 | f ′ ( z ) | 1 + | f ( z ) | 2 ≤ C . {\displaystyle {\frac {2|f'(z)|}{1+|f(z)|^{2}}}\leq C.}
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.
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