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
is a normal family , if each sequence in a compact convergent subsequence (automatically continuous) with limit function contains.
So it has each sequence in a partial sequence with
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 .
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.
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