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

Given a set S and functions ${\displaystyle f_{n}:S\to \mathbb {C} }$  (or to any normed vector space), the series

${\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}$ 

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

${\displaystyle \sum _{n=0}^{\infty }\|f_{n}\|:=\sum _{n=0}^{\infty }\sup _{x\in S}|f_{n}(x)|<\infty .}$ 

Normal convergence implies uniform absolute convergence, i.e., uniform convergence of the series of nonnegative functions ${\displaystyle \sum _{n=0}^{\infty }|f_{n}(x)|}$ ; this fact is essentially the Weierstrass M-test. However, they should not be confused; to illustrate this, consider

${\displaystyle f_{n}(x)={\begin{cases}1/n,&x=n,\\0,&x\neq n.\end{cases}}}$ 

Then the series ${\displaystyle \sum _{n=0}^{\infty }|f_{n}(x)|}$  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).

Local normal convergence edit

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

${\displaystyle \sum _{n=0}^{\infty }f_{n}\mid _{U}}$ 

is normally convergent, i.e. such that

${\displaystyle \sum _{n=0}^{\infty }\|f_{n}\|_{U}<\infty }$ 

where the norm ${\displaystyle \|\cdot \|_{U}}$  is the supremum over the domain U.

Compact normal convergence edit

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

${\displaystyle \sum _{n=0}^{\infty }f_{n}\mid _{K}}$ 

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.

• 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 ${\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}$  is normally convergent to ${\displaystyle f}$ , then any re-arrangement of the sequence (ƒ₁, ƒ₂, ƒ₃ ...) also converges normally to the same ƒ. That is, for every bijection ${\displaystyle \tau :\mathbb {N} \to \mathbb {N} }$ , ${\displaystyle \sum _{n=0}^{\infty }f_{\tau (n)}(x)}$  is normally convergent to ${\displaystyle f}$ .

• Modes of convergence (annotated index)