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


 * $$f_n(x) = \begin{cases} 1/n, & x = n, \\ 0, & x \ne n. \end{cases}$$

Then the series $$\sum_{n=0}^\infty |f_n(x)|$$ is uniformly convergent (for any &epsilon; take n ≥ 1/&epsilon;), 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).

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 &fnof;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 ƒn 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 (ƒ1, ƒ2, ƒ3 ...) 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 $$f$$.