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)