# Montel theorem

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; 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).

Montel's theorem on conditions for normality of a family of holomorphic functions (principle of normality, see [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, 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

 [1] P. Montel, "Leçons sur les séries de polynomes à une variable complexe" , Gauthier-Villars (1910) [2] P. Montel, "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars (1927)