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In [[mathematics]], with special application to [[complex analysis]], a '''normal family''' is a [[relatively compact subspace|pre-compact]] family of [[continuous function]]s. Informally, this means that the functions in the family are not exceedingly numerous or widely spread out; rather, they stick together in a relatively "compact" manner. It is of general interest to understand compact sets in [[function space]]s, since these are usually truly infinite-dimensional in nature.
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More formally, a [[indexed family|family]] (equivalently, a [[set (mathematics)|set]]) ''F'' of continuous functions ''f'' defined on some [[complete metric space]] ''X'' with values in another complete metric space ''Y'' is called '''normal''' 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''.
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==Complex analysis==
This definition is often used in complex analysis for spaces of [[holomorphic function]]s. 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. Thus in complex analysis a normal family ''F'' of holomorphic functions in a region ''X'' of the complex plane with values in ''Y'' = '''C''' is such that every sequence in 'F' contains a subsequence which converges uniformly on compact subsets of ''X'' to a holomorphic function. [[Montel's theorem]] asserts that every locally bounded family of holomorphic functions is normal.
 
Another space where this is often used is the space of [[meromorphic function]]s. This is similar to the holomorphic case, but instead of using the standard [[metric space|metric]] (distance) for convergence we must use the [[Riemann sphere|spherical metric]]. That is if ''d'' is the spherical metric, then want
:<math>f_n(z) \to f(z)</math>  
compactly to mean that  
 
:<math>d\left(f_n(z),f(z)\right)\, </math>
 
goes to 0 uniformly on compact subsets.
 
==Naming==
[[Paul Montel]] coined the term "normal family" in 1912.<ref>{{cite book
| url = http://books.google.ca/books?id=BHc2b0iCoy8C
| title = Classical Topics in Complex Function Theory
| author = Reinhold Remmert, Leslie Kay
| publisher = Springer
| year = 1998
| pages = 154
| accessdate = 2009-03-01
}}</ref>
 
Note that this is a classical definition that, while very often used, is not really consistent with modern naming. In more modern language, one would give a metric on the space of continuous (holomorphic) functions that corresponds to convergence on compact subsets and then you would say "precompact set of functions" in such a metric space instead of saying "normal family of continuous (holomorphic) functions". This added generality however makes it more cumbersome to use since one would need to define the metric mentioned above.
 
==Criteria==
* [[Montel's theorem|1'st criterion by Montel]]
* [http://eom.springer.de/n/n067510.htm 2'nd criterion by Montel]
* 3rd criterion by Marty.<ref>[http://eom.springer.de/n/n067510.htm  The Online Encyclopaedia of Mathematics. Edited by [[Michiel Hazewinkel]]. Description of Marty's criterion, but without giving a name]</ref> Marty proved criterion of normality for families of [[meromorphic function]]s in the case of one [[complex number|complex]] variable.<ref>[http://www.math.md/files/basm/y2005-n2/y2005-n2-(pp94-98).pdf N. Gashitsoi : On a criterion of normality for mappings. ] [http://www.math.md/publications/basm/ BASM n.2 (48), 2005]</ref><ref>[http://pcwww.liv.ac.uk/~lrempe/workshops/liv_jan_08/ Lecture 1 : An Introduction to Holomorphic Dynamics. I. Introduction; Normal Families by L. Rempe. Department of Mathematical Sciences, University of Liverpool Liverpool, January 2008]</ref>
 
==See also==
*[[Fundamental normality test]]
 
==Notes==
{{reflist}}
 
==References==
*{{cite book | author = John B. Conway | title = Functions of One Complex Variable I | publisher = Springer-Verlag | year = 1978 | isbn=0-387-90328-3 }}
*{{cite book | author = J. L. Schiff | title = Normal Families | publisher = Springer-Verlag | year = 1993 | isbn=0-387-97967-0 }}
* [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.
 
{{PlanetMath attribution|id=5753|title=normal family}}
 
[[Category:Continuous mappings]]
[[Category:Topology of function spaces]]
[[Category:Complex analysis]]

Latest revision as of 09:45, 7 March 2014

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