Vitali's theorem (function theory)

The set of Vitali , named after the Italian mathematician Giuseppe Vitali (1875-1932), is a statement of function theory . It is a sufficient criterion for the compact convergence of a sequence of holomorphic functions .

statement

A domain and a sequence of holomorphic functions are given , which fulfill the following two conditions: ${\displaystyle G\subset \mathbb {C} }$ ${\displaystyle f_{n}\colon G\to \mathbb {C} }$

• Let the sequence be locally restricted, i.e. That is , for every compact subset of there exists a number such that ${\displaystyle \left(f_{n}\right)_{n\in \mathbb {N} }}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle m_{K}>0}$

${\displaystyle |f_{n}(z)|\leq m_{K}}$for everyone and ${\displaystyle z\in K}$ ${\displaystyle n\in \mathbb {N} }$

applies.

• There is a subset of with at least one accumulation point in , so that the limit value ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle G}$

${\displaystyle \lim \limits _{n\to \infty }f_{n}(z)}$

for each exists. ${\displaystyle z\in A}$

Under these assumptions, the compact convergence of the sequence in applies . ${\displaystyle \left(f_{n}\right)_{n\in \mathbb {N} }}$ ${\displaystyle G}$

literature

• Wolfgang Fischer, Ingo Lieb: Function theory . 7th edition. Vieweg 1994, ISBN 3-528-67247-1

Web links

• Eric W. Weisstein : Vitali's Convergence Theorem . In: MathWorld (English).