Normal family

Inmathematics,with special application tocomplex analysis,anormal familyis apre-compactsubset of the space ofcontinuous functions.Informally, this means that thefunctionsin 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 familyFsatisfies a particular property (e.g. isholomorphic), then the property also holds for eachlimit pointof the setF.

More formally, letXandYbetopological spaces.The set of continuous functions$f:X\to Y$has a naturaltopologycalled thecompact-open topology.Anormal familyis apre-compactsubset with respect to this topology.

IfYis ametric space,then the compact-open topology is equivalent to the topology ofcompact convergence,and we obtain a definition which is closer to the classical one: A collectionFof continuous functions is called anormal family if everysequenceof functions inFcontains asubsequencewhichconverges uniformly on compact subsetsofXto a continuous function fromXtoY.That is, for every sequence of functions inF,there is a subsequence$f_{n}(x)$and a continuous function$f(x)$fromXtoYsuch that the following holds for everycompactsubsetKcontained inX:

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

where$d_{Y}$is themetricofY.

Normal families of holomorphic functions

The concept arose incomplex analysis,that is the study ofholomorphic functions.In this case,Xis anopen subsetof thecomplex plane,Yis the complex plane, and the metric onYis given by$d_{Y}(y_{1},y_{2})=|y_{1}-y_{2}|$.As a consequence ofCauchy's integral theorem,a sequence of holomorphic functions that converges uniformly on compact sets must converge to a holomorphic function. That is, eachlimit pointof a normal family is holomorphic.

Normal families of holomorphic functions provide the quickest way of proving theRiemann mapping theorem.

More generally, if the spacesXandYareRiemann surfaces,andYis equipped with the metric coming from theuniformization theorem,then each limit point of a normal family of holomorphic functions$f:X\to Y$is also holomorphic.

For example, ifYis theRiemann sphere,then the metric of uniformization is thespherical distance.In this case, a holomorphic function fromXtoYis called ameromorphic 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 theoremstates that a family of locally bounded holomorphic functions is normal. TheMontel-Caratheodorytheorem states that the family of meromorphic functions that omit three distinct values in theextended complex planeis 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$F$of meromorphic functions from adomain$U\subset \mathbb {C}$to the complex plane is a normal family if and only if for each compact subsetKofUthere exists a constantCso that for each$f\in F$and eachzinKwe have

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

Indeed, the expression on the left is the formula for thepull-backof thearclengthelement on theRiemann sphereto the complex plane via the inverse ofstereographic projection.

History

Paul Montelfirst coined the term "normal family" in 1911. 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 phrasepre-compact subsetmight 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

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
• Ahlfors, Lars V.(1978),Complex analysis. An introduction to the theory of analytic functions of one complex variable,International Series in Pure and Applied Mathematics (3rd ed.), McGraw-Hill,ISBN0070006571
• Beardon, Alan F.(1979),Complex analysis.The argument principle in analysis and topology,John Wiley & Sons,ISBN0471996718
• Chuang, Chi Tai (1993),Normal families of meromorphic functions,World Scientific,ISBN9810212577
• Conway, John B. (1978).Functions of One Complex Variable I.Springer-Verlag.ISBN0-387-90328-3.
• Gamelin, Theodore W. (2001).Complex analysis.Springer-Verlag.ISBN0-387-95093-1.
• 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.ISBN0-13-181629-2.
• Schiff, J. L. (1993).Normal Families.Springer-Verlag.ISBN0-387-97967-0.

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