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 "S" and functions $f\_n\; :\; X\; o\; mathbb\{C\}$ (or to any normed vector space), the series:$sum\_\{n=0\}^\{infty\}\; f\_n(x)$is called uniformly absolutely-convergent if the series of nonnegative functions :$sum\_\{n=0\}^\{infty\}\; |f\_n(x)|$is uniformly convergent. ^{[http://books.google.com/books?id=azS2ktxrz3EC&pg=PA1648&lpg=PA1648&dq=%22uniformly+absolutely+convergent%22&source=web&ots=MYazWvxtmR&sig=aw_SN9-AUjm36Jfn-W3evKiQOp4&hl=en&sa=X&oi=book_result&resnum=2&ct=result#PPA1647,M1]}

Distinctions

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", 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.

ee also

*Modes of convergence (annotated index)