- Uniform absolute-convergence
mathematics, uniform absolute-convergence is a type of convergencefor series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.
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 convergentseries 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.
Given a set "S" and functions (or to any
normed vector space), the series:is called uniformly absolutely-convergent if the series of nonnegative functions :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]
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.
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.
* 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.
Modes of convergence (annotated index)
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