130
II - Convergence: Discrete variables
Since we can reason in this way about I_as about 1+, we see that each of
these sums taken over 1+ and 1_ converges unconditionally. The converse
being obvious, one thus obtains the following result:
Theorem 7. For a series Eu(i), i E I, to converge unconditionally it is
necessary and sufficient that the series E lu( i) 1 converges unconditionally,
i. e. that there is a number M Z 0 such that
(15.6)
I: lu(i)1 So M
iEF
for every finite subset F of I. One then has
(15.7)
II: u(i)1 So I: lu(i)l·
To establish the inequality (7), one uses the fact that for every r > 0 there
exists a finite subset F of I such that the sum s(F) is equal to the total sum
s(I) of u(i) to within r; it follows that
Is(I)1 So Is(F)1 + r So I: lu(i)1 + r So I: lu(i)1 + r,
iEF
iEI
whence the result.
This theorem shows that, for unconditional convergence, there is no difference between convergence (in short) and absolute convergence: this is the
great difference from the classical concept. For this reason, one often speaks
of absolutely summable instead of unconditionally convergent families, and,
in practice, one says only that the series u( i) is absolutely or commutatively
convergent; the terminology depends on the author. Although there are important series which do not fit into this scheme - one meets them for example
in the theory of Fourier series of a single variable, without speaking of alternating series Ii la Leibniz -, unconditional convergence suffices in the great
majority of cases because it is the "discrete" analogue of, and much simpler
than, the modern theory of integration, which one might define as a theory of unconditional convergence for "continuous" sums, i.e. indexed by the
points of an interval of lR. or something analogous, a cube in lR. 3 for example. These two theories allow one to calculate in a quasi-algebraic way, and
one uses hardly any other nowadays in analysis for the reason that simple
ordered series, and even less, semi-convergent series, appear only very rarely
in dimensions greater than 1. The Riemann series in two variables m and n
studied in nO 12 is typical in this respect.
Corollary (Cauchy-Schwarz inequality for series). Let (Ui) and (Vi)
be two families of complex numbers such that the series L IUil 2 and L IVil2
converge unconditionally. Then so does the series L UiVi, and
II - Convergence: Discrete variables
Since we can reason in this way about I_as about 1+, we see that each of
these sums taken over 1+ and 1_ converges unconditionally. The converse
being obvious, one thus obtains the following result:
Theorem 7. For a series Eu(i), i E I, to converge unconditionally it is
necessary and sufficient that the series E lu( i) 1 converges unconditionally,
i. e. that there is a number M Z 0 such that
(15.6)
I: lu(i)1 So M
iEF
for every finite subset F of I. One then has
(15.7)
II: u(i)1 So I: lu(i)l·
To establish the inequality (7), one uses the fact that for every r > 0 there
exists a finite subset F of I such that the sum s(F) is equal to the total sum
s(I) of u(i) to within r; it follows that
Is(I)1 So Is(F)1 + r So I: lu(i)1 + r So I: lu(i)1 + r,
iEF
iEI
whence the result.
This theorem shows that, for unconditional convergence, there is no difference between convergence (in short) and absolute convergence: this is the
great difference from the classical concept. For this reason, one often speaks
of absolutely summable instead of unconditionally convergent families, and,
in practice, one says only that the series u( i) is absolutely or commutatively
convergent; the terminology depends on the author. Although there are important series which do not fit into this scheme - one meets them for example
in the theory of Fourier series of a single variable, without speaking of alternating series Ii la Leibniz -, unconditional convergence suffices in the great
majority of cases because it is the "discrete" analogue of, and much simpler
than, the modern theory of integration, which one might define as a theory of unconditional convergence for "continuous" sums, i.e. indexed by the
points of an interval of lR. or something analogous, a cube in lR. 3 for example. These two theories allow one to calculate in a quasi-algebraic way, and
one uses hardly any other nowadays in analysis for the reason that simple
ordered series, and even less, semi-convergent series, appear only very rarely
in dimensions greater than 1. The Riemann series in two variables m and n
studied in nO 12 is typical in this respect.
Corollary (Cauchy-Schwarz inequality for series). Let (Ui) and (Vi)
be two families of complex numbers such that the series L IUil 2 and L IVil2
converge unconditionally. Then so does the series L UiVi, and