Normal family

In mathematics , normal families are particularly important in function theory and complex dynamics .

general definition

Be and full metric spaces . An amount of continuous functions ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle F}$

${\displaystyle f\colon X\to Y}$

is a normal family , if each sequence in a compact convergent subsequence (automatically continuous) with limit function contains. ${\displaystyle F}$ ${\displaystyle f_{\infty }}$

So it has each sequence in a partial sequence with ${\displaystyle (f_{n})_{n}}$ ${\displaystyle F}$ ${\displaystyle (f_{n_{k}})_{k}}$

${\displaystyle \lim _{k\to \infty }\sup _{x\in K}d_{Y}(f_{n_{k}}(x),f_{\infty }(x))=0}$

for all compact subsets . ${\displaystyle K\subset X}$

Normal families in function theory

In function theory, one generally chooses a domain in the domain of definition and the Riemann number sphere as the target space , provided with the chordal metric. It follows from Weierstraß's theorem of convergence that the limit value of a compact convergent sequence of holomorphic functions is again holomorphic or constant . ${\displaystyle X}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle Y}$ ${\displaystyle \infty }$

Montel's Little Theorem says that a locally uniformly bounded family of holomorphic functions is normal. After the huge set of Montel a family of analytic functions is normal when there with there, so no function of the family one of the values or accept. ${\displaystyle a,b\in \mathbb {C} }$ ${\displaystyle a\neq b}$ ${\displaystyle a}$ ${\displaystyle b}$

literature

• Paul Montel : Sur les familles normales des fonctions analytiques , Ann. ENS 33, 233-302 (1916).
• Joel Schiff : Normal families . Springer, 1993, ISBN 978-1-4612-0907-2
• Reinhold Remmert , Georg Schumacher : Function Theory 2. 3. Edition. Springer, 2007, ISBN 3540404325