Editing Normal family

In [[mathematics]], with special application to [[complex analysis]], a ''normal family'' is a [[relatively compact subspace|pre-compact]] subset of the space of [[continuous function (topology)|continuous function]]s. Informally, this means that the [[function (mathematics)|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 space]]s. The set of continuous functions  <math>f: X \to Y</math> has a natural [[topology]] called the [[compact-open topology]]. A '''normal family''' is a [[relatively compact subspace|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]],<ref>{{cite book
| author = Munkres
| title = Topology, Theorem 46.8.
}}</ref> 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 [[compact convergence|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 <math>f_n(x)</math> and a continuous function <math>f(x)</math> from ''X'' to ''Y'' such that the following holds for every [[compact space|compact]] subset ''K'' contained in ''X'':

:<math>\lim_{n\rightarrow\infty} \sup_{x\in K} d_Y(f_n(x),f(x)) = 0</math>

where <math>d_Y</math> is the [[metric (mathematics)|metric]] of ''Y''.

==Normal families of holomorphic functions==
The concept arose in [[complex analysis]], that is the study of [[holomorphic function]]s. 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 <math>d_Y(y_1,y_2) = |y_1-y_2|</math>.  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]].<ref>See for example
*{{harvnb|Ahlfors|1953}}, {{harvnb|Ahlfors|1966}}, {{harvnb|Ahlfors|1978}}
*{{harvnb|Conway|1978}}
*{{harvnb|Beardon|1979}}</ref>

More generally, if the spaces ''X'' and ''Y'' are [[Riemann surface]]s, and ''Y'' is equipped with the metric coming from the [[uniformization theorem]], then each limit point of  a normal family of holomorphic functions <math> f: X \to Y</math> 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's theorem|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.

[[Frédéric Marty|Marty's theorem]]<ref>
{{cite book
| author = Gamelin
| title = Complex Analysis, Section 12.1.
}}</ref>
provides a criterion equivalent to normality in the context of meromorphic functions: A family <math>F</math> of meromorphic functions from a [[Domain (mathematical analysis)|domain]] <math> U \subset \mathbb{C} </math> 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 <math> f \in F </math> and each ''z'' in ''K'' we have

:<math> \frac{2|f'(z)|}{1 + |f(z)|^2} \leq C. </math>

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]].

==History==
[[Paul Montel]] first coined the term "normal family" in 1911.<ref>P. Montel, C. R. Acad. Sci. Paris 153 (1911), 996–998; Jahrbuch '''42''', page 426</ref><ref>{{cite book |last=Remmert |first=Rienhard |url=https://books.google.com/books?id=BHc2b0iCoy8C |title=Classical Topics in Complex Function Theory |publisher=Springer |year=1998 |pages=154 |isbn=9780387982212 |translator=[[Leslie M. Kay]] |authorlink=Reinhold Remmert |accessdate=2009-03-01}}</ref>
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==
{{reflist|30em}}

==References==
*{{citation|last=Ahlfors|first= Lars V.|authorlink=Lars Ahlfors|title=Complex analysis. An introduction to the theory of analytic functions of one complex variable|publisher= McGraw-Hill|year= 1953}}
*{{citation|last=Ahlfors|first= Lars V.|authorlink=Lars Ahlfors|title=Complex analysis. An introduction to the theory of analytic functions of one complex variable|edition=2nd|series= International Series in Pure and Applied Mathematics|publisher= McGraw-Hill|year= 1966}}
*{{citation|last=Ahlfors|first= Lars V.|authorlink=Lars Ahlfors|title=Complex analysis. An introduction to the theory of analytic functions of one complex variable|edition=3rd|series= International Series in Pure and Applied Mathematics|publisher= McGraw-Hill|year= 1978|isbn= 0070006571}}
*{{citation|last=Beardon|first= Alan F.|authorlink=Alan Frank Beardon|title=Complex analysis.The argument principle in analysis and topology|publisher= John Wiley & Sons|year= 1979|isbn= 0471996718}}
*{{citation|last=Chuang|first= Chi Tai|title=Normal families of meromorphic functions|publisher=World Scientific| year= 1993|isbn= 9810212577}}
*{{cite book | first=  John B.|last= Conway | title = Functions of One Complex Variable I | publisher = Springer-Verlag | year = 1978 | isbn=0-387-90328-3 }}
*{{cite book | first=  Theodore W.|last= Gamelin | title = Complex analysis | publisher = Springer-Verlag | year = 2001 | isbn=0-387-95093-1}}
* [http://genealogy.math.ndsu.nodak.edu/id.php?id=130208 Marty, Frederic] :  Recherches sur la répartition des valeurs d’une function méromorphe. Ann. Fac. Sci. Univ. Toulouse, 1931, 28, N 3, p.&nbsp;183–261.
*{{citation|first=Paul|last=Montel|authorlink=Paul Montel|title=Leçons sur les familles normales de fonctions analytiques et leur applications|language=fr|publisher=Gauthier-Villars |year= 1927}}
*{{cite book | first=  James R.|last= Munkres | title = Topology | publisher = Prentice Hall | year = 2000 | isbn=0-13-181629-2 }} 
*{{cite book | first = J. L.|last= Schiff | title = Normal Families | publisher = Springer-Verlag | year = 1993 | isbn=0-387-97967-0 }}

{{PlanetMath attribution|id=5753|title=normal family}}

[[Category:Theory of continuous functions]]
[[Category:Topology of function spaces]]
[[Category:Complex analysis]]