mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, uniform absolute-convergence is a type of

convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Wei ...

for

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

s. 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 In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...

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

f_n : X \to \mathbb

(or to any

normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...

), the series :

\sum_^ f_n(x)

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

\sum_^ , f_n(x),

is uniformly convergent.Kiyosi Itō (1987). ''Encyclopedic Dictionary of Mathematics'', ''MIT Press''.
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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''/''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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

, these notions are equivalent.

Properties

* If a series of functions into ''C'' (or any

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

) 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.

References