Normal family

In

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

of Y.

Normal families of holomorphic functions

The concept arose in

, Y is the complex plane, and the metric on Y is given by ${\displaystyle d_{Y}(y_{1},y_{2})=|y_{1}-y_{2}|}$. 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 of a normal family is holomorphic.

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 ${\displaystyle f:X\to Y}$ is also holomorphic.

For example, if Y is the

, 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:

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^[3] provides a criterion equivalent to normality in the context of meromorphic functions: A family ${\displaystyle F}$ of meromorphic functions from a domain ${\displaystyle U\subset \mathbb {C} }$ 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 ${\displaystyle f\in F}$ and each z in K we have

${\displaystyle {\frac {2|f'(z)|}{1+|f(z)|^{2}}}\leq C.}$

Indeed, the expression on the left is the formula for the

.

History

Paul Montel first coined the term "normal family" in 1911.^[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.

See also

• Fundamental normality test

Notes

References

• Ahlfors, Lars V. (1953), Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill
• Ahlfors, Lars V. (1966), Complex analysis. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics (2nd ed.), McGraw-Hill
• ISBN 0070006571
• ISBN 0471996718
• Chuang, Chi Tai (1993), Normal families of meromorphic functions, World Scientific,
  ISBN 9810212577
• Conway, John B. (1978). Functions of One Complex Variable I. Springer-Verlag.
  ISBN 0-387-90328-3
  .
• Gamelin, Theodore W. (2001). Complex analysis. Springer-Verlag.
  ISBN 0-387-95093-1
  .
• Marty, Frederic : Recherches sur la répartition des valeurs d’une function méromorphe. Ann. Fac. Sci. Univ. Toulouse, 1931, 28, N 3, p. 183–261.
• Montel, Paul (1927), Leçons sur les familles normales de fonctions analytiques et leur applications (in French), Gauthier-Villars
• Munkres, James R. (2000). Topology. Prentice Hall.
  ISBN 0-13-181629-2
  .
• Schiff, J. L. (1993). Normal Families. Springer-Verlag.
  ISBN 0-387-97967-0
  .

This article incorporates material from normal family on

Creative Commons Attribution/Share-Alike License

.