Normal convergence  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. 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 formula_1 (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, i.e.,  Normal convergence implies, but should not be confused with, uniform absolute convergence, i.e. uniform convergence of the series of nonnegative functions formula_4. To illustrate this, consider  Then the series formula_4 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 "ƒ""n" restricted to the domain "U" is normally convergent, i.e. such that where the norm formula_9 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 "ƒ""n" 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.