Dini's theorem

In mathematics , Dini's theorem (named after Ulisse Dini ) says that a monotonic sequence of real-valued continuous functions with continuous limit functions converges uniformly on compacts .

statement

Are a compact topological space , ${\displaystyle X}$

${\displaystyle (f_{i}\colon X\rightarrow \mathbb {R} )_{i\in \mathbb {N} }}$

a sequence of real-valued, continuous functions with

${\displaystyle f_{i}(x)\leq f_{i+1}(x)}$

for all natural numbers and all and there is a continuous limit function , that is ${\displaystyle i}$ ${\displaystyle x\in X}$ ${\displaystyle f}$

${\displaystyle \lim _{i\to \infty }f_{i}(x)=f(x)}$

for all , then the sequence already converges equally against , that is ${\displaystyle x\in X}$ ${\displaystyle f}$

${\displaystyle \lim _{i\to \infty }\sup _{x\in X}|f_{i}(x)-f(x)|=0.}$

proof

For a given set ${\displaystyle \varepsilon >0}$

${\displaystyle E_{i}:=\{x\in X\mid f(x)-f_{i}(x)<\varepsilon \}}$.

Since the sequence of converges to pointwise , they form an overlap of , which is open because of the assumed continuity. The coverage increases monotonically because the sequence of functions has this property. Because is compact, it is already covered by a finite number of . If the largest index of these finitely many coverage sets is true for all larger indices . So is ${\displaystyle f_{i}}$ ${\displaystyle f}$ ${\displaystyle E_{i}}$ ${\displaystyle X}$ ${\displaystyle (E_{i})_{i}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle E_{i}}$ ${\displaystyle N}$ ${\displaystyle E_{i}=X}$ ${\displaystyle i}$

${\displaystyle |f(x)-f_{i}(x)|=f(x)-f_{i}(x)<\varepsilon \,}$for everyone and , ${\displaystyle x\in X}$ ${\displaystyle i>N\,}$

from which the claim follows.

comment

Dini's theorem also applies to monotonically decreasing sequences, as can be seen either by a correspondingly adapted proof or by transition to the sequence . ${\displaystyle (-f_{i})_{i\in \mathbb {N} }}$

The requirement that the limit function is continuous again cannot be dispensed with, as can be seen in the example on . ${\displaystyle f_{i}(x)=1-x^{i}}$ ${\displaystyle X=[0,1]}$

literature

• Otto Forster : Analysis. Volume 3: Measure and integration theory, integral theorems in R ⁿ and applications , 8th improved edition. Springer Spectrum, Wiesbaden, 2017, ISBN 978-3-658-16745-5 .
• Dirk Werner : Functional Analysis . Springer, Berlin 2005, ISBN 3-540-43586-7 .