Normal convergence

In mathematics , the concept of normal convergence is used to characterize infinite series of functions . The term was introduced by the French mathematician René Louis Baire .

definition

Let be any topological space . For functions and any subset of be ${\displaystyle X}$ ${\displaystyle f\colon X\to \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle X}$

${\displaystyle \Vert f\Vert _{A}:=\sup _{x\in A}\left|f(x)\right|}$

the supreme norm . A series of functions is called normally convergent if there is a neighborhood of for each such that: ${\displaystyle \textstyle \sum _{n=0}^{\infty }f_{n}}$ ${\displaystyle f_{n}\colon X\to \mathbb {C} }$ ${\displaystyle x\in X}$ ${\displaystyle U}$ ${\displaystyle x}$

${\displaystyle \sum _{n=0}^{\infty }\Vert f_{n}\Vert _{U}<\infty }$

example

Observe the sequence of functions on the compact interval with . Then it's our turn ${\displaystyle f_{n}(x):=x^{n}}$ ${\displaystyle I:=[-q,q]}$ ${\displaystyle 0<q<1}$ ${\displaystyle \Vert f_{n}\Vert _{I}=q^{n}}$

${\displaystyle \sum _{n=0}^{\infty }\Vert f_{n}\Vert _{I}=\sum _{n=0}^{\infty }q^{n}}$

converges (as a geometric series because of ). The series of functions is thus normally convergent and its limit function is continuously on . ${\displaystyle |q|<1}$ ${\displaystyle \textstyle x\mapsto {\frac {1}{1-x}}}$ ${\displaystyle I}$

properties

The concept of normal convergence is a relatively strong concept of convergence, because for every series that converges in normal, it is also locally uniformly convergent there , that is, for every point there is a neighborhood in which the series converges evenly . Thus every normally convergent series is also compactly convergent , since this follows from the locally uniform convergence. ${\displaystyle X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle U(x_{0})}$

The following facts are also important:

• Linear combinations and the product of normally convergent series are normally convergent again.

• If all are continuous, then the limit function is also continuous if it converges normally. ${\displaystyle f_{n}}$ ${\displaystyle \textstyle \sum _{n=0}^{\infty }f_{n}}$ ${\displaystyle \textstyle \sum _{n=0}^{\infty }f_{n}}$

• If a series converges normally, then all rearrangements of this series converge normally, to the same limit function.

literature

• R. Remmert: Function theory I. Springer Verlag, Berlin / Heidelberg / New York 1989, ISBN 3-540-51238-1 .
• Konrad Königsberger : Analysis 1. Springer Verlag, 3rd edition, 1995.