Normal convergence

Definition

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

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{f}_n(x)}$

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

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

Distinctions

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

${\displaystyle f_n(x)=\{\begin{array}{ll}1/n,& x=n,\\ 0,& x\ne n.\end{array}}$

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

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{f}_n{\mid }_U}$

is normally convergent, i.e. such that

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\Vert f_n{\Vert }_U<\mathrm{\infty }}$

where the norm ${\displaystyle \Vert \cdot {\Vert }_U}$ 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

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{f}_n{\mid }_K}$

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 ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{f}_n(x)}$ 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 ${\displaystyle \tau :\mathbb{N}\to \mathbb{N}}$, ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{f}_{\tau (n)}(x)}$ is normally convergent to ${\displaystyle f}$.