Uniform absolute-convergence


In mathematics, uniform absolute-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.

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 , the series
is called uniformly absolutely-convergent if the series of nonnegative functions
is uniformly convergent.

Distinctions

A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒn = xn/n on the open interval, then the series Σfn converges uniformly by comparison of the partial sums to those of Σn/n, and the series Σ|fn| converges absolutely at each point by the geometric series test, but Σ|fn| 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 absolutely-convergent. If the topological space is locally compact, these notions are equivalent.

Properties