Motivation
A convergent series of numbers can often be reordered in such a way that the new series diverges. This is not possible for series of nonnegative numbers, however, so the notion of absolute-convergence precludes this phenomenon. When dealing with uniformly convergent series of functions, the same phenomenon occurs: the series can potentially be reordered into a non-uniformly convergent series, or a series which does not even converge pointwise. This is impossible for series of nonnegative functions, so the notion of uniform absolute-convergence can be used to rule out these possibilities.Definition
Given a set ''X'' and functions (or to anyDistinctions
A series can be uniformly convergent ''and'' absolutely convergent without being uniformly absolutely-convergent. For example, if ''ƒ''''n''(''x'') = ''x''''n''/''n'' on the open interval (−1,0), then the series Σ''f''''n''(''x'') converges uniformly by comparison of the partial sums to those of Σ(−1)''n''/''n'', and the series Σ, ''f''''n''(''x''), converges absolutely ''at each point'' by the geometric series test, but Σ, ''f''''n''(''x''), does not converge uniformly. Intuitively, this is because the absolute-convergence gets slower and slower as ''x'' approaches −1, where convergence holds but absolute convergence fails.Generalizations
If a series of functions is uniformly absolutely-convergent on some neighborhood of each point of a topological space, it is locally uniformly absolutely-convergent. If a series is uniformly absolutely-convergent on all compact subsets of a topological space, it is compactly (uniformly) absolutely-convergent. If the topological space is locally compact, these notions are equivalent.Properties
* If a series of functions into ''C'' (or any Banach space) is uniformly absolutely-convergent, then it is uniformly convergent. * Uniform absolute-convergence is independent of the ordering of a series. This is because, for a series of nonnegative functions, uniform convergence is equivalent to the property that, for any ε > 0, there are finitely many terms of the series such that excluding these terms results in a series with total sum less than the constant function ε, and this property does not refer to the ordering.See also
*References