More precisely, a real or complex-valued series is said to converge absolutely if
The vital importance of absolute convergence in the study of infinite series rests in the following two principles. On the one hand absolute convergence is strong enough that such series retain certain basic properties of finite sums — the most important ones being rearrangement of the terms and convergence of products of two infinite series — that are unfortunately not possessed by all convergent series. On the other hand absolute convergence is weak enough to occur very often in practice. Indeed, in some (though not all) branches of mathematics in which series are applied, the existence of convergent but not absolutely convergent series is little more than a curiosity.
One may study the convergence of series whose terms are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm:
A norm on an abelian group G (written additively, with identity element 0) is a real-valued function on G such that:
(i) The norm of the identity element of G is zero:
(ii) The norm of any nonidentity element is strictly positive:
(iii) For every x in G,
(iv) For every x, y in G,
Then the function induces on G the structure of a metric space (and in particular, a topology). We can therefore consider G-valued series and define such a series to be absolutely convergent if
If the metric d on G is complete, then every absolutely convergent series is convergent. The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergence — a series is convergent if and only if its tails can be made arbitrarily small in norm — and apply the triangle inequality.
In particular, for series with values in any Banach space, absolute convergence implies convergence.
Of course a series may be convergent without being absolutely convergent, the standard example being the alternating harmonic series. However, many standard tests which show that a series is convergent in fact show absolute convergence, notably the ratio and root tests. This has the important consequence that a power series is absolutely convergent on the interior of its disk of convergence.
It is standard in calculus courses to say that a real series which is convergent but not absolutely convergent is conditionally convergent. However, in the more general context of G-valued series a distinction is made between absolute and unconditional convergence, and the assertion that a real or complex series which is not absolutely convergent is necessarily conditionally convergent (i.e., not unconditionally convergent) is then a theorem, not a definition. This is discussed in more detail below.
Given a series with values in a normed abelian group G and a permutation of the natural numbers, one builds a new series , said to be a rearrangement of the original series. A series is said to be unconditionally convergent if all rearrangements of the series are convergent to the same value.
When G is complete, absolute convergence implies unconditional convergence. (Again, the proof requires little more than applying the Cauchy criterion and then the triangle inequality.)
The issue of the converse is much more interesting. For real series it follows from the Riemann rearrangement theorem that unconditional convergence implies absolute convergence. Since a series with values in a finite-dimensional normed space is absolutely convergent iff each of its one-dimensional projections is absolutely convergent, it follows easily that absolute and unconditional convergence coincide for -valued series.
But there is an unconditionally and nonabsolutely convergent series with values in Hilbert space : if is an orthonormal basis, take .
Remarkably, a theorem of Dvoretzky-Rogers asserts that every infinite-dimensional Banach space admits an unconditionally but nonabsolutely convergent series.
The Cauchy product of two series converges to the product of the sums if at least one of the series converges absolutely. That is, suppose:
The Cauchy product is defined as the sum of terms where:
Then, if either the or sum converges absolutely, then
The integral of a real or complex-valued function is said to converge absolutely if One also says that is absolutely integrable.
When is a closed bounded interval, every continuous function is integrable, and since continuous implies continuous, similarly every continuous function is absolutely integrable. It is not generally true that absolutely integrable functions on are integrable: let be a nonmeasurable subset and take , where is the characteristic function of S. Then f is not Lebesgue measurable but |f| is constant. However, it is a standard result that if f is Riemann integrable, so is |f|. This holds also for the Lebesgue integral; see below. On the other hand a function f may be Kurzweil-Henstock integrable (or "gauge integrable") while |f| is not. This includes the case of improperly Riemann integrable functions.
Similarly, when A is an interval of infinite length it is well-known that there are improperly Riemann integrable functions f which are not absolutely integrable. Indeed, given any series one can consider the associated step function defined by . Then converges absolutely, converges conditionally or diverges according to the corresponding behavior of .
Another example of a convergent but not absolutely convergent improper Riemann integral is .
On any measure space A the Lebesgue integral of a real-valued function is defined in terms of its positive and negative parts, so the facts:
(i) f integrable implies |f| integrable
(ii) f measurable, |f| integrable implies f integrable
are essentially built into the definition of the Lebesgue integral. In particular, applying the theory to the counting measure on a set S, one recovers the notion of unordered summation of series developed by Moore-Smith using (what are now called) nets. When is the set of natural numbers, Lebesgue integrability, unordered summability and absolute convergence all coincide.
Finally, all of the above holds for integrals with values in a Banach space. The definition of a Banach-valued Riemann integral is an evident modification of the usual one. For the Lebesgue integral one needs to circumvent the decomposition into positive and negative parts with Daniell's more functional analytic approach, obtaining the Bochner integral.
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