Normal convergence

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In mathematics normal convergence is a type of convergence for series of functions.

A convergent series of numbers can often be reordered in such a way that the new series diverges. A stronger form of convergence, namely absolute convergence, eliminates this phenomenon. When dealing with uniformly convergent series of functions, the same phenomenon occurs: the series can potentially be reordered into a non- uniformly convergent series. The notion of normal convergence rules out this possibility.

1 History
2 Definition
3 Generalizations
- 3.1 Local normal convergence
- 3.2 Compact normal convergence
4 Properties

[edit] History

The concept of normal convergence was first introduced by René Baire in 1908 in his book Leçons sur les théories générales de l'analyse.

[edit] Definition

Given a topological space X and functions $f_n : X \to \mathbb{C}$ (or to any normed vector space), the series

$\sum_{n=0}^{\infty} f_n$

is called normally convergent if it is "uniformly absolutely convergent", i.e., if the series of nonnegative functions

$\sum_{n=0}^{\infty} |f_n|$

is uniformly convergent (where, as usual, $| f n | (x): = | f n (x) |$ ).

This condition is equivalent to requiring that the the series of uniform norms of the functions converges^[1], i.e.,

$\sum_{n=0}^{\infty} \|f_n\| < \infty$ ,

where $\|f_n\| := \sup_X |f_n(x)|$ (hence the term "normal").

Note: This is a strictly stronger property than being "uniformly and absolutely convergent" (uniformly convergent on X and absolutely convergent at each point of X).

[edit] Generalizations

[edit] Local normal convergence

A series can be called "locally normally convergent on X" if each point x in X has a neighborhood U such that the series of restrictions

$\sum_{n=0}^{\infty} f_{n\mid U}$

is normally convergent, i.e. such that

$\sum_{n=0}^{\infty} \|f_{n\mid U}\| < \infty$ .

[edit] Compact normal convergence

A series is said to be "normally convergent on compact subsets of X" or "compactly normally convergent on X" if for every compact subset K of X, the series of restrictions

$\sum_{n=0}^{\infty} f_{n\mid K}$

is normally convergent.

Note: if X is locally compact (even in the weakest sense), local normal convergence and compact normal convergence are equivalent.

[edit] Properties

Every normal convergent series is uniformly convergent, locally uniformly convergent, and compactly uniformly convergent.
If $\sum_{n=0}^{\infty} f_n(x)$ is normal convergent to $f$ , then for every bijection $\tau: \mathbb{N} \to \mathbb{N}$ , $\sum_{n=0}^{\infty} f_{\tau(n)}(x)$ converges normally to $f$