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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====
<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>