Universal variable formulation: Difference between revisions
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In [[mathematics]], '''uniform absolute-convergence''' is a type of [[Convergent series|convergence]] for [[series (mathematics)|series]] of [[function (mathematics)|function]]s. Like [[absolute convergence|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|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 <math>f_n : X \to \mathbb{C}</math> (or to any [[normed vector space]]), the series | |||
:<math>\sum_{n=0}^{\infty} f_n(x)</math> | |||
is called '''uniformly absolutely-convergent''' if the series of nonnegative functions | |||
:<math>\sum_{n=0}^{\infty} |f_n(x)|</math> | |||
is uniformly convergent.<ref>[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 Kiyosi Itō (1987). ''Encyclopedic Dictionary of Mathematics'', ''MIT Press''.]</ref> | |||
== Distinctions == | |||
A series can be uniformly convergent ''and'' absolutely convergent without being uniformly absolutely-convergent. For example, if ''ƒ''<sub>''n''</sub>(''x'') = ''x''<sup>''n''</sup>/''n'' on the open interval (−1,0), then the series Σ''f''<sub>''n''</sub>(''x'') converges uniformly by comparison of the partial sums to those of Σ(−1)<sup>''n''</sup>/''n'', and the series Σ|''f''<sub>''n''</sub>(''x'')| converges absolutely ''at each point'' by the geometric series test, but Σ|''f''<sub>''n''</sub>(''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== | |||
*[[Modes of convergence (annotated index)]] | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Uniform Absolute-Convergence}} | |||
[[Category:Mathematical analysis]] | |||
[[Category:Convergence (mathematics)]] | |||
Latest revision as of 18:55, 7 March 2013
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 (or to any normed vector space), the series
is called uniformly absolutely-convergent if the series of nonnegative functions
is uniformly convergent.[1]
Distinctions
A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒn(x) = xn/n on the open interval (−1,0), then the series Σfn(x) converges uniformly by comparison of the partial sums to those of Σ(−1)n/n, and the series Σ|fn(x)| converges absolutely at each point by the geometric series test, but Σ|fn(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.