Compactness principle

in the theory of functions of a complex variable

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

Let $H_D$ be the complex vector space of holomorphic functions in a domain $D$ of the space $\mathbf C^n$, $n\geq1$, with the topology of uniform convergence on compacta $K\subset D$. The compactness principle can be stated in a more abstract form: A closed set $\Phi\subset H_D$ is compact in $H_D$ if and only if it is bounded in $H_D$. 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)