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.
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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
is called normally convergent if the series of uniform norms of the terms of the series converges, 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
Then the series
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 iff 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 fn restricted to the domain U
is normally convergent, i.e. such that
where the norm
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 fn 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.