# 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 (or to any normed vector space), the series

is called **normally convergent** if the series of uniform norms of the terms of the series converges,[1] i.e.,

## Distinctions

Normal convergence implies, but should not be confused with, uniform absolute convergence, i.e. uniform convergence of the series of nonnegative functions . To illustrate this, consider

Then the series 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*

is normally convergent, i.e. such that

where the norm 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*

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 is normally convergent to , 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 , is normally convergent to .