The simple-uniform convergence is a concept of convergence from the mathematical branch of analysis . It is a weakening of the uniform convergence . The term was defined by Ulisse Dini, among others .
definition
Be a subset. A sequence of functions that converges pointwise is said to converge to simply-uniformly if
D.
f
⊂
R.
{\ displaystyle D_ {f} \ subset \ mathbb {R}}
(
f
n
:
D.
f
→
R.
)
n
∈
N
{\ displaystyle (f_ {n} \ colon D_ {f} \ to \ mathbb {R}) _ {n \ in \ mathbb {N}}}
f
{\ displaystyle f}
∀
ε
>
0
∃
N
∈
N
∀
x
∈
D.
f
:
Card
(
{
n
∣
n
≥
N
,
|
f
n
(
x
)
-
f
(
x
)
|
<
ε
}
)
=
ℵ
0
{\ displaystyle \ forall \ varepsilon> 0 \ \ exists N \ in \ mathbb {N} \ \ forall x \ in D_ {f}: \ operatorname {Card} (\ {n \ mid n \ geq N, \ \ left | f_ {n} (x) -f (x) \ right | <\ varepsilon \}) = \ aleph _ {0}}
applies. With is meant the power of .
ℵ
0
{\ displaystyle \ aleph _ {0}}
N
{\ displaystyle \ mathbb {N}}
properties
Every uniformly convergent sequence of functions is also simply-uniformly convergent.
Individual evidence
^ EW Hobson: The Theory of Functions of a Real Variable and the Theory of Fourier's Series. 2nd edition, Cambridge, ISBN 978-1418186517 , pp. 105-106.
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">