Normal convergence

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, but should not be confused with, uniform absolute convergence, i.e. uniform convergence of the series of nonnegative functions ${\displaystyle \sum _{n=0}^{\infty }|f_{n}(x)|}$. 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).

• 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)