Normal convergence
From Wikipedia, the free encyclopedia
From Wikipedia, the free encyclopedia
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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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 (or to any normed vector space), the series
is called normally convergent if the series of uniform norms of the terms of the series converges,[1] i.e.,
Normal convergence implies uniform absolute convergence, i.e., uniform convergence of the series of nonnegative functions ; this fact is essentially the Weierstrass M-test. However, they should not be confused; to illustrate this, consider
Then the series 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).
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 ƒn restricted to the domain U
is normally convergent, i.e. such that
where the norm is the supremum over the domain U.
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
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.
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