Point-by-point convergence

The pointwise convergence is in the analysis a concept of convergence for sequences of functions . A sequence of functions converges pointwise to a function if for all points ( "Points") from the common domain the result against converges. ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f,}$ ${\displaystyle x}$ ${\displaystyle (f_{n}(x))_{n\in \mathbb {N} }}$ ${\displaystyle f(x)}$

definition

Consider a sequence of functions , . The sequence of functions is called pointwise convergent to a function if holds for all ${\displaystyle f_{n}\colon D\to \mathbb {R} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle x\in D}$

${\displaystyle \lim _{n\rightarrow \infty }f_{n}(x)=f(x)}$.

Then you write

${\displaystyle \lim _{n\to \infty }f_{n}=f{\text{ punktweise}}}$

or

${\displaystyle f_{n}{\xrightarrow {\text{pktw.}}}f\ (n\to \infty )}$.

Formally then converges pointwise to if and only if ${\displaystyle f_{n}}$ ${\displaystyle f}$

${\displaystyle \forall x\in D\ \forall \varepsilon >0\ \exists N\in \mathbb {N} \ \forall n\geq N\colon \left|f_{n}(x)-f(x)\right|<\varepsilon }$,

that is, it must be for each and every natural number type, such that for all the following applies: . ${\displaystyle x}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N}$ ${\displaystyle n\geq N}$ ${\displaystyle \left|f_{n}(x)-f(x)\right|<\varepsilon }$

example

For example, the sequence converges with ${\displaystyle f_{n}}$

${\displaystyle f_{n}\colon x\mapsto x^{n}}$

in the interval point by point against the function with ${\displaystyle [0,1]}$ ${\displaystyle f\colon [0,1]\to \mathbb {R} }$

${\displaystyle f(x)={\begin{cases}0,&0\leq x<1,\\1,&x=1\end{cases}}}$

because obviously applies

${\displaystyle \lim _{n\to \infty }x^{n}=0\ \mathrm {f{\ddot {u}}r\ alle} \ x\in [0,1){\text{ und }}\lim _{n\to \infty }1^{n}=1.}$

Demarcation

It should be noted, however, that point-wise convergence is not synonymous with uniform convergence , since z. For example, the above example converges point by point, but in no way converges uniformly (so every term of the sequence is continuously differentiable everywhere, but the limit function is not even continuous): Uniform convergence is a much stronger statement.

A weakening of the point-wise convergence is the point-wise convergence μ-almost everywhere .

For pointwise convergence, the values ​​of the functions do not necessarily have to be real numbers; they can be elements of any topological space . ${\displaystyle f_{n}}$

See also

Uniform convergence

literature

• Konrad Königsberger : Analysis 1 . Springer, Berlin 2004, ISBN 3-540-41282-4