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Convergence of a series
 
Convergence of a series
  
<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/n/n067/n067430/n0674301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= \sum _ { k= } 1 ^  \infty  u _ {k}  $$
  
formed by bounded mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n0674302.png" /> from a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n0674303.png" /> into a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n0674304.png" />, such that the series with positive terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n0674305.png" /> formed by the norms of the mappings,
+
formed by bounded mappings $  u _ {k} : X \rightarrow Y $
 +
from a set $  X $
 +
into a normed space $  Y $,  
 +
such that the series with positive terms $  \sum _ {k=} 1  ^  \infty  \| u _ {k} \| $
 +
formed by the norms of the mappings,
  
<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/n/n067/n067430/n0674306.png" /></td> </tr></table>
+
$$
 +
\| u _ {k} \|  = \
 +
\sup \{ {\| u _ {k} ( x) \| } : {x \in X } \}
 +
,
 +
$$
  
 
converges.
 
converges.
  
Normal convergence of the series (1) implies absolute and uniform convergence of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n0674307.png" /> consisting of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n0674308.png" />; the converse is not true. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n0674309.png" /> is the real-valued function defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743013.png" />, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743014.png" /> converges absolutely, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743015.png" /> diverges.
+
Normal convergence of the series (1) implies absolute and uniform convergence of the series $  \sum _ {k=} 1  ^  \infty  u _ {k} ( x) $
 +
consisting of elements of $  Y $;  
 +
the converse is not true. For example, if $  u _ {k} : \mathbf R \rightarrow \mathbf R $
 +
is the real-valued function defined by $  u _ {k} ( x) = ( \sin  \pi x ) / k $
 +
for $  k \leq  x \leq  k + 1 $
 +
and $  u _ {k} ( x) = 0 $
 +
for $  x \in \mathbf R \setminus  [ k, k+ 1] $,  
 +
then the series $  \sum _ {k=} 1  ^  \infty  u _ {k} ( x) $
 +
converges absolutely, whereas $  \sum _ {k=} 1  ^  \infty  \| u _ {k} \| = \sum _ {k=} 1  ^  \infty  1 / k $
 +
diverges.
  
Suppose, in particular, that each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743016.png" /> is a piecewise-continuous function on a non-compact interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743017.png" /> and that (1) converges normally. Then one can integrate term-by-term on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743018.png" />:
+
Suppose, in particular, that each $  u _ {k} : \mathbf R \rightarrow Y $
 +
is a piecewise-continuous function on a non-compact interval $  I \subset  \mathbf R $
 +
and that (1) converges normally. Then one can integrate term-by-term on $  I $:
  
<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/n/n067/n067430/n06743019.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { I } f ( t)  d t  = \
 +
\sum _ { k= } 1 ^  \infty  \int\limits _ { I } u _ {k} ( t)  d t .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743021.png" /> is an interval, have left and right limits at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743022.png" />. Then the improper integral
+
Let $  f: I \times A \rightarrow Y $,  
 +
where $  I \subset  \mathbf R $
 +
is an interval, have left and right limits at each point of $  I $.  
 +
Then the improper integral
  
<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/n/n067/n067430/n06743023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { I } f ( t ; \lambda ) d t ,\ \
 +
\lambda \in A ,
 +
$$
  
is called normally convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743024.png" /> if there exists a piecewise-continuous positive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743025.png" /> such that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743026.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743027.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743028.png" />; and 2) the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067430/n06743029.png" /> converges. Normal convergence of (2) implies its absolute and uniform convergence; the converse is not true.
+
is called normally convergent on $  A $
 +
if there exists a piecewise-continuous positive function $  g : \mathbf R \rightarrow \mathbf R $
 +
such that: 1) $  \| f( x ;  \lambda ) \| \leq  g ( x) $
 +
for any $  x \in I $
 +
and any $  \lambda \in A $;  
 +
and 2) the integral $  \int _ {I} g ( t)  d t $
 +
converges. Normal convergence of (2) implies its absolute and uniform convergence; the converse is not true.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Functions of a real variable" , Addison-Wesley  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Schwartz,  "Cours d'analyse" , '''1''' , Hermann  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Functions of a real variable" , Addison-Wesley  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Schwartz,  "Cours d'analyse" , '''1''' , Hermann  (1967)</TD></TR></table>

Revision as of 08:03, 6 June 2020


Convergence of a series

$$ \tag{1 } f = \sum _ { k= } 1 ^ \infty u _ {k} $$

formed by bounded mappings $ u _ {k} : X \rightarrow Y $ from a set $ X $ into a normed space $ Y $, such that the series with positive terms $ \sum _ {k=} 1 ^ \infty \| u _ {k} \| $ formed by the norms of the mappings,

$$ \| u _ {k} \| = \ \sup \{ {\| u _ {k} ( x) \| } : {x \in X } \} , $$

converges.

Normal convergence of the series (1) implies absolute and uniform convergence of the series $ \sum _ {k=} 1 ^ \infty u _ {k} ( x) $ consisting of elements of $ Y $; the converse is not true. For example, if $ u _ {k} : \mathbf R \rightarrow \mathbf R $ is the real-valued function defined by $ u _ {k} ( x) = ( \sin \pi x ) / k $ for $ k \leq x \leq k + 1 $ and $ u _ {k} ( x) = 0 $ for $ x \in \mathbf R \setminus [ k, k+ 1] $, then the series $ \sum _ {k=} 1 ^ \infty u _ {k} ( x) $ converges absolutely, whereas $ \sum _ {k=} 1 ^ \infty \| u _ {k} \| = \sum _ {k=} 1 ^ \infty 1 / k $ diverges.

Suppose, in particular, that each $ u _ {k} : \mathbf R \rightarrow Y $ is a piecewise-continuous function on a non-compact interval $ I \subset \mathbf R $ and that (1) converges normally. Then one can integrate term-by-term on $ I $:

$$ \int\limits _ { I } f ( t) d t = \ \sum _ { k= } 1 ^ \infty \int\limits _ { I } u _ {k} ( t) d t . $$

Let $ f: I \times A \rightarrow Y $, where $ I \subset \mathbf R $ is an interval, have left and right limits at each point of $ I $. Then the improper integral

$$ \tag{2 } \int\limits _ { I } f ( t ; \lambda ) d t ,\ \ \lambda \in A , $$

is called normally convergent on $ A $ if there exists a piecewise-continuous positive function $ g : \mathbf R \rightarrow \mathbf R $ such that: 1) $ \| f( x ; \lambda ) \| \leq g ( x) $ for any $ x \in I $ and any $ \lambda \in A $; and 2) the integral $ \int _ {I} g ( t) d t $ converges. Normal convergence of (2) implies its absolute and uniform convergence; the converse is not true.

References

[1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[2] N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) (Translated from French)
[3] L. Schwartz, "Cours d'analyse" , 1 , Hermann (1967)
How to Cite This Entry:
Normal convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_convergence&oldid=17565
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article