Compactness principle

in the theory of functions of a complex variable

The condition of compactness of families of analytic functions. An infinite family of holomorphic functions in a domain of the complex -plane is called compact if one can select from any sequence a subsequence converging to an analytic function in or, what is the same, converging uniformly in the interior of , that is, uniformly converging on any compactum . The compactness principle was formulated by P. Montel in 1927 (see [1]): In order that a family be compact, it is necessary and sufficient that it be uniformly bounded in the interior of , that is, uniformly bounded on any compactum .

Let be the complex vector space of holomorphic functions in a domain of the space , , with the topology of uniform convergence on compacta . The compactness principle can be stated in a more abstract form: A closed set is compact in if and only if it is bounded in . The notion of a compact family of analytic functions is closely related to that of a normal family. See also Vitali theorem.

References

[1]	P. Montel, "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars (1927)
[2]	B. Malgrange, "Lectures on the theory of functions of several complex variables" , Tata Inst. (1958)

References

[a1]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) pp. Sect. 86 (Translated from Russian)