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''of a sequence of functions (mappings)''
 
''of a sequence of functions (mappings)''
  
A property of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952301.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952302.png" /> is an arbitrary set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952303.png" /> is a metric space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952304.png" /> converging to a function (mapping) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952305.png" />, requiring that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952306.png" /> there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952307.png" /> (independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952308.png" />) such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u0952309.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523010.png" /> the inequality
+
A property of a sequence $  f _ {n} : X \rightarrow Y $,  
 +
where $  X $
 +
is an arbitrary set, $  Y $
 +
is a metric space, $  n = 1, 2 \dots $
 +
converging to a function (mapping) $  f: X \rightarrow Y $,  
 +
requiring that for every $  \epsilon > 0 $
 +
there is a number $  n _  \epsilon  $(
 +
independent of $  x $)  
 +
such that for all $  n > n _  \epsilon  $
 +
and all $  x \in X $
 +
the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523011.png" /></td> </tr></table>
+
$$
 +
\rho ( f ( x), f _ {n} ( x))  < \epsilon
 +
$$
  
 
holds. This is equivalent to
 
holds. This is equivalent to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523012.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty } \
 +
\sup _ {x \in X } \
 +
\rho ( f _ {n} ( x), f ( x))  = 0.
 +
$$
  
In order that a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523013.png" /> converges uniformly on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523014.png" /> to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523015.png" /> it is necessary and sufficient that there is a sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523017.png" />, as well as a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523018.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523019.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523020.png" /> the inequality
+
In order that a sequence $  \{ f _ {n} \} $
 +
converges uniformly on a set $  X $
 +
to a function $  f $
 +
it is necessary and sufficient that there is a sequence of numbers $  \{ \alpha _ {n} \} $
 +
such that $  \lim\limits _ {n \rightarrow \infty }  \alpha _ {n} = 0 $,  
 +
as well as a number $  n _ {0} $
 +
such that for $  n > n _ {0} $
 +
and all $  x \in X $
 +
the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523021.png" /></td> </tr></table>
+
$$
 +
\rho ( f _ {n} ( x), f ( x))  \leq  \alpha _ {n}  $$
  
 
holds.
 
holds.
  
Example. The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523023.png" /> converges uniformly on any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523025.png" />, but does not converge uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523026.png" />.
+
Example. The sequence $  \{ f _ {n} ( x) \} = \{ x  ^ {n} \} $,
 +
$  n = 1, 2 \dots $
 +
converges uniformly on any interval $  [ 0, a] $,
 +
$  0 < a < 1 $,  
 +
but does not converge uniformly on $  [ 0, 1] $.
  
 
A necessary and sufficient condition for uniform convergence that does not use the limit function is given by the [[Cauchy criteria|Cauchy criterion]] for uniform convergence.
 
A necessary and sufficient condition for uniform convergence that does not use the limit function is given by the [[Cauchy criteria|Cauchy criterion]] for uniform convergence.
Line 21: Line 62:
 
==Properties of uniformly-convergent sequences.==
 
==Properties of uniformly-convergent sequences.==
  
 +
1) If  $  Y $
 +
is a normed linear space and two sequences of mappings  $  f _ {n} :  X \rightarrow Y $
 +
and  $  g _ {n} :  X \rightarrow Y $
 +
converge uniformly on  $  X $,
 +
then for any  $  \lambda , \mu \in \mathbf C $
 +
the sequence  $  \{ \lambda f _ {n} + \mu g _ {n} \} $
 +
also converges uniformly on  $  X $.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523027.png" /> is a normed linear space and two sequences of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523029.png" /> converge uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523030.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523031.png" /> the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523032.png" /> also converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523033.png" />.
+
2) If $  Y $
 +
is a linear normed ring, if the sequence  $  f _ {n} : X \rightarrow Y $,
 +
$  n = 1, 2 \dots $
 +
is uniformly convergent on $  X $
 +
and if  $  g: X \rightarrow Y $
 +
is a bounded mapping, then the sequence $  \{ gf _ {n} \} $
 +
also converges uniformly on $  X $.
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523034.png" /> is a linear normed ring, if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523036.png" /> is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523037.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523038.png" /> is a bounded mapping, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523039.png" /> also converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523040.png" />.
+
3) If $  X $
 +
is a topological space, $  Y $
 +
is a metric space and if a sequence of mappings  $  f _ {n} : X \rightarrow Y $,  
 +
continuous at  $  x _ {0} \in X $,
 +
converges uniformly on $  X $
 +
to  $  f: X \rightarrow Y $,  
 +
then $  f $
 +
is also continuous at  $  x _ {0} $,
 +
that is,
  
3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523041.png" /> is a topological space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523042.png" /> is a metric space and if a sequence of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523043.png" />, continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523044.png" />, converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523047.png" /> is also continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523048.png" />, that is,
+
$$
 +
\lim\limits _ {x \rightarrow x _ {0} } \
 +
\lim\limits _ {n \rightarrow \infty } \
 +
f _ {n} ( x) = \
 +
\lim\limits _ {n \rightarrow \infty } \
 +
f _ {n} ( x _ {0} )  = \
 +
\lim\limits _ {n \rightarrow \infty } \
 +
\lim\limits _ {x \rightarrow x _ {0} } \
 +
f _ {n} ( x).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523049.png" /></td> </tr></table>
+
The condition of uniform convergence of the sequence  $  \{ f _ {n} \} $
 +
on  $  X $
 +
is essential in this result, in the sense that there are sequences of numerical functions, continuous on an interval, that converge at all points to a function that is not continuous on the interval in question. An example is  $  f _ {n} ( x) = x  ^ {n} $,
 +
$  n = 1, 2 \dots $
 +
on  $  [ 0, 1] $.  
 +
Uniform convergence of a sequence of continuous functions is not a necessary condition for continuity of the limit function. However, if  $  X $
 +
is a compact set,  $  Y $
 +
is the set of real numbers  $  \mathbf R $
 +
and if all functions in a sequence of continuous functions  $  f _ {n} :  X \rightarrow \mathbf R $
 +
simultaneously increase or decrease at all points  $  x \in X $
 +
and the sequence has a finite limit:
  
The condition of uniform convergence of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523051.png" /> is essential in this result, in the sense that there are sequences of numerical functions, continuous on an interval, that converge at all points to a function that is not continuous on the interval in question. An example is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523053.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523054.png" />. Uniform convergence of a sequence of continuous functions is not a necessary condition for continuity of the limit function. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523055.png" /> is a compact set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523056.png" /> is the set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523057.png" /> and if all functions in a sequence of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523058.png" /> simultaneously increase or decrease at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523059.png" /> and the sequence has a finite limit:
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  f _ {n} ( x)  = f ( x),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523060.png" /></td> </tr></table>
+
then in order that  $  f $
 +
be continuous on  $  X $
 +
it is necessary and sufficient that  $  \{ f _ {n} \} $
 +
converges uniformly on that set. Necessary, and simultaneously sufficient, conditions for the continuity of the limit of a sequence of continuous functions in general are given in terms of [[Quasi-uniform convergence|quasi-uniform convergence]] of the sequence.
  
then in order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523061.png" /> be continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523062.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523063.png" /> converges uniformly on that set. Necessary, and simultaneously sufficient, conditions for the continuity of the limit of a sequence of continuous functions in general are given in terms of [[Quasi-uniform convergence|quasi-uniform convergence]] of the sequence.
+
4) If a sequence of Riemann- (Lebesgue-) integrable functions  $  f _ {n} : [ a, b] \rightarrow \mathbf R $,
 +
$  n = 1, 2 \dots $
 +
converges uniformly on $  [ a, b] $
 +
to a function  $  f:  [ a, b] \rightarrow \mathbf R $,
 +
then this function is also Riemann- (respectively, Lebesgue-) integrable, for any  $  x \in [ a, b] $
 +
one has
  
4) If a sequence of Riemann- (Lebesgue-) integrable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523065.png" /> converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523066.png" /> to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523067.png" />, then this function is also Riemann- (respectively, Lebesgue-) integrable, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523068.png" /> one has
+
$$ \tag{* }
 +
\lim\limits _ {n \rightarrow \infty } \
 +
\int\limits _ { a } ^ { x }  f _ {n} ( t) dt  = \
 +
\int\limits _ { a } ^ { x }  f ( t) dt  = \
 +
\int\limits _ { a } ^ { x }
 +
\lim\limits _ {n \rightarrow \infty }  f _ {n} ( t) dt,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
and the convergence of the sequence  $  \{ \int _ {a}  ^ {x} f _ {n} ( t)  dt \} $
 +
to  $  \int _ {a}  ^ {x} f ( t)  dt $
 +
is uniform on  $  [ a, b] $.  
 +
Formula (*) has been generalized to the case of a [[Stieltjes integral|Stieltjes integral]]. If, however, a sequence of integrable functions  $  f _ {n} $,
 +
$  n = 1, 2 \dots $
 +
on  $  [ a, b] $
 +
only converges at each point of the interval to an integrable function  $  f $,
 +
then (*) need not hold.
  
and the convergence of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523070.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523071.png" /> is uniform on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523072.png" />. Formula (*) has been generalized to the case of a [[Stieltjes integral|Stieltjes integral]]. If, however, a sequence of integrable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523074.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523075.png" /> only converges at each point of the interval to an integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523076.png" />, then (*) need not hold.
+
5) If a sequence of continuously differentiable functions $  f _ {n} : [ a, b] \rightarrow \mathbf R $,  
 +
$  n = 1, 2 \dots $
 +
on $  [ a, b] $
 +
converges at some point $  x _ {0} \in [ a, b] $
 +
and if the sequence of derivatives  $  \{ df _ {n} /dx \} $
 +
converges uniformly on  $  [ a, b] $,  
 +
then the sequence  $  \{ f _ {n} \} $
 +
also converges uniformly on  $  [ a, b] $,
 +
its limit is a continuously differentiable function on the interval and
  
5) If a sequence of continuously differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523078.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523079.png" /> converges at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523080.png" /> and if the sequence of derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523081.png" /> converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523082.png" />, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523083.png" /> also converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523084.png" />, its limit is a continuously differentiable function on the interval and
+
$$
 +
{
 +
\frac{d}{dx }
 +
}
 +
\lim\limits _ {n \rightarrow \infty }  f _ {n} ( x) = \
 +
\lim\limits _ {n \rightarrow \infty } \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523085.png" /></td> </tr></table>
+
\frac{df _ {n} ( x) }{dx }
 +
,\ \
 +
a \leq  x \leq  b.
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523086.png" /> be a set and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523087.png" /> a metric space. A family of functions (mappings) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523089.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523090.png" /> a topological space, is said to be uniformly convergent as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523091.png" /> to the function (mapping) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523092.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523093.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523094.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523095.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523097.png" /> the inequality
+
Let $  X $
 +
be a set and $  Y $
 +
a metric space. A family of functions (mappings) $  f _  \alpha  : X \rightarrow Y $,  
 +
$  \alpha \in \mathfrak U $,  
 +
with $  \mathfrak U $
 +
a topological space, is said to be uniformly convergent as $  \alpha \rightarrow \alpha _ {0} \in \mathfrak U $
 +
to the function (mapping) $  f: X \rightarrow Y $
 +
if for every $  \epsilon > 0 $
 +
there is a neighbourhood $  U ( \alpha _ {0} ) $
 +
of $  \alpha _ {0} $
 +
such that for all $  \alpha \in U( \alpha _ {0} ) $
 +
and $  x \in X $
 +
the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523098.png" /></td> </tr></table>
+
$$
 +
\rho ( f( x), f _  \alpha  ( x))  < \epsilon
 +
$$
  
 
holds.
 
holds.
Line 54: Line 188:
 
For uniformly-convergent families of functions there are properties similar to the above-mentioned properties of uniformly-convergent sequences of functions.
 
For uniformly-convergent families of functions there are properties similar to the above-mentioned properties of uniformly-convergent sequences of functions.
  
The concept of uniform convergence of mappings can be generalized to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u09523099.png" /> is a [[Uniform space|uniform space]], in particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095230/u095230100.png" /> is a topological group.
+
The concept of uniform convergence of mappings can be generalized to the case when $  Y $
 +
is a [[Uniform space|uniform space]], in particular, when $  Y $
 +
is a topological group.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The theorem that a monotone sequence of continuous functions converges uniformly to its pointwise limit if this limit is continuous, is known as Dini's theorem.
+
The theorem that a monotone sequence of continuous functions converges uniformly to its [[pointwise limit]] if this limit is continuous, is known as [[Dini theorem|Dini's theorem]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR></table>

Latest revision as of 08:27, 6 June 2020


of a sequence of functions (mappings)

A property of a sequence $ f _ {n} : X \rightarrow Y $, where $ X $ is an arbitrary set, $ Y $ is a metric space, $ n = 1, 2 \dots $ converging to a function (mapping) $ f: X \rightarrow Y $, requiring that for every $ \epsilon > 0 $ there is a number $ n _ \epsilon $( independent of $ x $) such that for all $ n > n _ \epsilon $ and all $ x \in X $ the inequality

$$ \rho ( f ( x), f _ {n} ( x)) < \epsilon $$

holds. This is equivalent to

$$ \lim\limits _ {n \rightarrow \infty } \ \sup _ {x \in X } \ \rho ( f _ {n} ( x), f ( x)) = 0. $$

In order that a sequence $ \{ f _ {n} \} $ converges uniformly on a set $ X $ to a function $ f $ it is necessary and sufficient that there is a sequence of numbers $ \{ \alpha _ {n} \} $ such that $ \lim\limits _ {n \rightarrow \infty } \alpha _ {n} = 0 $, as well as a number $ n _ {0} $ such that for $ n > n _ {0} $ and all $ x \in X $ the inequality

$$ \rho ( f _ {n} ( x), f ( x)) \leq \alpha _ {n} $$

holds.

Example. The sequence $ \{ f _ {n} ( x) \} = \{ x ^ {n} \} $, $ n = 1, 2 \dots $ converges uniformly on any interval $ [ 0, a] $, $ 0 < a < 1 $, but does not converge uniformly on $ [ 0, 1] $.

A necessary and sufficient condition for uniform convergence that does not use the limit function is given by the Cauchy criterion for uniform convergence.

Properties of uniformly-convergent sequences.

1) If $ Y $ is a normed linear space and two sequences of mappings $ f _ {n} : X \rightarrow Y $ and $ g _ {n} : X \rightarrow Y $ converge uniformly on $ X $, then for any $ \lambda , \mu \in \mathbf C $ the sequence $ \{ \lambda f _ {n} + \mu g _ {n} \} $ also converges uniformly on $ X $.

2) If $ Y $ is a linear normed ring, if the sequence $ f _ {n} : X \rightarrow Y $, $ n = 1, 2 \dots $ is uniformly convergent on $ X $ and if $ g: X \rightarrow Y $ is a bounded mapping, then the sequence $ \{ gf _ {n} \} $ also converges uniformly on $ X $.

3) If $ X $ is a topological space, $ Y $ is a metric space and if a sequence of mappings $ f _ {n} : X \rightarrow Y $, continuous at $ x _ {0} \in X $, converges uniformly on $ X $ to $ f: X \rightarrow Y $, then $ f $ is also continuous at $ x _ {0} $, that is,

$$ \lim\limits _ {x \rightarrow x _ {0} } \ \lim\limits _ {n \rightarrow \infty } \ f _ {n} ( x) = \ \lim\limits _ {n \rightarrow \infty } \ f _ {n} ( x _ {0} ) = \ \lim\limits _ {n \rightarrow \infty } \ \lim\limits _ {x \rightarrow x _ {0} } \ f _ {n} ( x). $$

The condition of uniform convergence of the sequence $ \{ f _ {n} \} $ on $ X $ is essential in this result, in the sense that there are sequences of numerical functions, continuous on an interval, that converge at all points to a function that is not continuous on the interval in question. An example is $ f _ {n} ( x) = x ^ {n} $, $ n = 1, 2 \dots $ on $ [ 0, 1] $. Uniform convergence of a sequence of continuous functions is not a necessary condition for continuity of the limit function. However, if $ X $ is a compact set, $ Y $ is the set of real numbers $ \mathbf R $ and if all functions in a sequence of continuous functions $ f _ {n} : X \rightarrow \mathbf R $ simultaneously increase or decrease at all points $ x \in X $ and the sequence has a finite limit:

$$ \lim\limits _ {n \rightarrow \infty } f _ {n} ( x) = f ( x), $$

then in order that $ f $ be continuous on $ X $ it is necessary and sufficient that $ \{ f _ {n} \} $ converges uniformly on that set. Necessary, and simultaneously sufficient, conditions for the continuity of the limit of a sequence of continuous functions in general are given in terms of quasi-uniform convergence of the sequence.

4) If a sequence of Riemann- (Lebesgue-) integrable functions $ f _ {n} : [ a, b] \rightarrow \mathbf R $, $ n = 1, 2 \dots $ converges uniformly on $ [ a, b] $ to a function $ f: [ a, b] \rightarrow \mathbf R $, then this function is also Riemann- (respectively, Lebesgue-) integrable, for any $ x \in [ a, b] $ one has

$$ \tag{* } \lim\limits _ {n \rightarrow \infty } \ \int\limits _ { a } ^ { x } f _ {n} ( t) dt = \ \int\limits _ { a } ^ { x } f ( t) dt = \ \int\limits _ { a } ^ { x } \lim\limits _ {n \rightarrow \infty } f _ {n} ( t) dt, $$

and the convergence of the sequence $ \{ \int _ {a} ^ {x} f _ {n} ( t) dt \} $ to $ \int _ {a} ^ {x} f ( t) dt $ is uniform on $ [ a, b] $. Formula (*) has been generalized to the case of a Stieltjes integral. If, however, a sequence of integrable functions $ f _ {n} $, $ n = 1, 2 \dots $ on $ [ a, b] $ only converges at each point of the interval to an integrable function $ f $, then (*) need not hold.

5) If a sequence of continuously differentiable functions $ f _ {n} : [ a, b] \rightarrow \mathbf R $, $ n = 1, 2 \dots $ on $ [ a, b] $ converges at some point $ x _ {0} \in [ a, b] $ and if the sequence of derivatives $ \{ df _ {n} /dx \} $ converges uniformly on $ [ a, b] $, then the sequence $ \{ f _ {n} \} $ also converges uniformly on $ [ a, b] $, its limit is a continuously differentiable function on the interval and

$$ { \frac{d}{dx } } \lim\limits _ {n \rightarrow \infty } f _ {n} ( x) = \ \lim\limits _ {n \rightarrow \infty } \ \frac{df _ {n} ( x) }{dx } ,\ \ a \leq x \leq b. $$

Let $ X $ be a set and $ Y $ a metric space. A family of functions (mappings) $ f _ \alpha : X \rightarrow Y $, $ \alpha \in \mathfrak U $, with $ \mathfrak U $ a topological space, is said to be uniformly convergent as $ \alpha \rightarrow \alpha _ {0} \in \mathfrak U $ to the function (mapping) $ f: X \rightarrow Y $ if for every $ \epsilon > 0 $ there is a neighbourhood $ U ( \alpha _ {0} ) $ of $ \alpha _ {0} $ such that for all $ \alpha \in U( \alpha _ {0} ) $ and $ x \in X $ the inequality

$$ \rho ( f( x), f _ \alpha ( x)) < \epsilon $$

holds.

For uniformly-convergent families of functions there are properties similar to the above-mentioned properties of uniformly-convergent sequences of functions.

The concept of uniform convergence of mappings can be generalized to the case when $ Y $ is a uniform space, in particular, when $ Y $ is a topological group.

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3] J.L. Kelley, "General topology" , Springer (1975)

Comments

The theorem that a monotone sequence of continuous functions converges uniformly to its pointwise limit if this limit is continuous, is known as Dini's theorem.

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
[a2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78
How to Cite This Entry:
Uniform convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_convergence&oldid=30905
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article