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Montel's theorem on the approximation of analytic functions by polynomials: If  $  D $
is an open set in the complex  $  z $-plane not containing  $  z = \infty $
and  $  f ( z) $
is a single-valued function, analytic at each point  $  z \in D $, 
then there is a sequence of polynomials  $  \{ P _ {n} ( z) \} $
converging to  $  f ( z) $
at each  $  z \in D $. 
This theorem is one of the basic results in the theory of [[Approximation of functions of a complex variable|approximation of functions of a complex variable]]; it was obtained by P. Montel .

Montel's theorem on compactness conditions for a family of holomorphic functions (principle of compactness, see ): Let  $  \Phi = \{ f ( z) \} $
be an infinite family of holomorphic functions in a domain  $  D $
of the complex  $  z $-plane, then  $  \Phi $
is pre-compact, that is, any subsequence  $  \{ f _ {k} ( z) \} \subset  \Phi $
has a subsequence converging uniformly on compact subsets of  $  D $, 
if  $  \Phi $
is uniformly bounded in  $  D $. 
This theorem can be generalized to a domain  $  D $
in  $  \mathbf C  ^ {n} $, 
$  n \geq  1 $ (see [[Compactness principle|Compactness principle]]).

Montel's theorem on conditions for normality of a family of holomorphic functions (principle of normality, see [[#References|[2]]]): Let  $  \Phi = \{ f ( z) \} $
be an infinite family of holomorphic functions in a domain  $  D $
of the complex  $  z $-plane. If there are two distinct values  $  a $
and  $  b $
that are not taken by any of the functions  $  f ( z) \in \Phi $, 
then  $  \Phi $
is a [[Normal family|normal family]], that is, any sequence  $  \{ f _ {k} ( z) \} \subset  \Phi $
has a sequence uniformly converging on compact subsets of  $  D $
to a holomorphic function or to  $  \infty $. 
The conditions of this theorem can be somewhat weakened: It suffices that all  $  f ( z) \in \Phi $
do not take one of the values, say  $  a $, 
and that the other value  $  b $
is taken at most  $  m $
times,  $  1 \leq  m < \infty $. 
This theorem can be generalized to a domain  $  D $
in  $  \mathbf C  ^ {n} $, 
$  n \geq  1 $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Montel,   "Leçons sur les séries de polynomes à une variable complexe" , Gauthier-Villars  (1910)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Montel,   "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars  (1927)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''3''', Sect. 11; 1, Sect. 86; 3, Sect. 50, Chelsea  (1977)  (Translated from Russian)</TD></TR></table>