Normal convergence

Convergence of a series

(1)

formed by bounded mappings from a set into a normed space , such that the series with positive terms formed by the norms of the mappings,

converges.

Normal convergence of the series (1) implies absolute and uniform convergence of the series consisting of elements of ; the converse is not true. For example, if is the real-valued function defined by for and for , then the series converges absolutely, whereas diverges.

Suppose, in particular, that each is a piecewise-continuous function on a non-compact interval and that (1) converges normally. Then one can integrate term-by-term on :

Let , where is an interval, have left and right limits at each point of . Then the improper integral

(2)

is called normally convergent on if there exists a piecewise-continuous positive function such that: 1) for any and any ; and 2) the integral 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)