Normal family

In mathematics, with special application to complex analysis, a normal family is a pre-compact family 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. It is of general interest to understand compact sets in function spaces, since these are usually truly infinite-dimensional in nature.

More formally, a family (equivalently, a 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 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 $f_n(x)$ and a continuous function $f(x)$ from X to Y such that the following holds for every compact subset K contained in X:

$\lim_{n\rightarrow\infty} \sup_{x\in K} d_Y(f_n(x),f(x)) = 0$

where $d_Y(y_1,y_2)$ is the distance metric associated with the complete metric space Y.

Complex analysis

This definition is often used in complex analysis for spaces of holomorphic functions. In this case, sets X and Y are regions in the complex plane, and $d_Y(y_1,y_2) = |y_1-y_2|$ . 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. 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 functions. This is similar to the holomorphic case, but instead of using the standard metric (distance) for convergence we must use the spherical metric. That is if d is the spherical metric, then want

f_n(z) \to f(z)

compactly to mean that

d\left(f_n(z),f(z)\right)\,

goes to 0 uniformly on compact subsets.

Naming

Paul Montel coined the term "normal family" in 1912.^[1]

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

1'st criterion by Montel
2'nd criterion by Montel
3rd criterion by Marty.^[2] Marty proved criterion of normality for families of meromorphic functions in the case of one complex variable.^[3]^[4]

Notes

↑ Reinhold Remmert, Leslie Kay (1998). Classical Topics in Complex Function Theory. Springer. p. 154. Retrieved 2009-03-01.
↑ The Online Encyclopaedia of Mathematics. Edited by Michiel Hazewinkel. Description of Marty's criterion, but without giving a name
↑ N. Gashitsoi : On a criterion of normality for mappings. BASM n.2 (48), 2005
↑ Lecture 1 : An Introduction to Holomorphic Dynamics. I. Introduction; Normal Families by L. Rempe. Department of Mathematical Sciences, University of Liverpool Liverpool, January 2008

References

John B. Conway (1978). Functions of One Complex Variable I. Springer-Verlag. ISBN 0-387-90328-3.
J. L. Schiff (1993). Normal Families. Springer-Verlag. ISBN 0-387-97967-0.
Marty Frederic : Recherches sur la répartition des valeurs d’une function méromorphe. Ann. Fac. Sci. Univ. Toulouse, 1931, 28, N 3, p. 183–261.

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