# Normal convergence

In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.

## 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.

## Definition

Given a set S and functions $f_{n}:S\to \mathbb {C}$ (or to any normed vector space), the series
    $\sum _{n=0}^{\infty }f_{n}(x)$
is called normally convergent if the series of uniform norms of the terms of the series converges, i.e.,
    $\sum _{n=0}^{\infty }\|f_{n}\|:=\sum _{n=0}^{\infty }\sup _{x\in S}|f_{n}(x)|<\infty .$

## Distinctions

Normal convergence implies uniform absolute convergence, i.e., uniform convergence of the series of nonnegative functions $\sum _{n=0}^{\infty }|f_{n}(x)|$; this fact is essentially the Weierstrass M-test. However, they should not be confused; to illustrate this, consider
    $f_{n}(x)={\begin{cases}1/n,&x=n,\\0,&x\neq n.\end{cases}}$
Then the series $\sum _{n=0}^{\infty }|f_{n}(x)|$ is uniformly convergent (for any ε take n ≥ 1/ε), but the series of uniform norms is the harmonic series and thus diverges. An example using continuous functions can be made by replacing these functions with bump functions of height 1/n and width 1 centered at each natural number n.
As well, normal convergence of a series is different from norm-topology convergence, i.e. convergence of the partial sum sequence in the topology induced by the uniform norm. Normal convergence implies norm-topology convergence if and only if the space of functions under consideration is complete with respect to the uniform norm. (The converse does not hold even for complete function spaces: for example, consider the harmonic series as a sequence of constant functions).

## Generalizations
### 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 functions ƒ$_n$ restricted to the domain U
    $\sum _{n=0}^{\infty }f_{n}\mid _{U}$
is normally convergent, i.e. such that
    $\sum _{n=0}^{\infty }\|f_{n}\|_{U}<\infty$
where the norm $\|\cdot \|_{U}$ is the supremum over the domain U.

### 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 functions ƒ$_n$ restricted to K
    $\sum _{n=0}^{\infty }f_{n}\mid _{K}$
is normally convergent on K.
Note: if X is locally compact (even in the weakest sense), local normal convergence and compact normal convergence are equivalent.

## Properties

* Every normal convergent series is uniformly convergent, locally uniformly convergent, and compactly uniformly convergent. This is very important, since it assures that any re-arrangement of the series, any derivatives or integrals of the series, and sums and products with other convergent series will converge to the "correct" value.
* If $\sum _{n=0}^{\infty }f_{n}(x)$ is normally convergent to $f$, then any re-arrangement of the sequence (ƒ$_1$, ƒ$_2$, ƒ$_3$ ...) also converges normally to the same ƒ. That is, for every bijection $\tau :\mathbb {N} \to \mathbb {N}$, $\sum _{n=0}^{\infty }f_{\tau (n)}(x)$ is normally convergent to $f$.