Normal convergence

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)