140
II - Convergence: Discrete variables
(18.1)
1= U I j
jEJ
of I into pairwise disjoint sets I j (so as to avoid repeating the terms of the
given family several times), indexed by a set J, which must be finite or countable like I and the Ij , and to calculate the sum s(I) of all the ~(i) by adding
the partial sums s(Ij) corresponding to the various sets I j . Associativity can
then be expressed as follows:
Theorem 13. Let (u(i)), i E I, be a family of complex numbers indexed
by a countable set I and let I = UIj, j E J, be a partition of I. For the
given family to converge unconditionally it is necessary and sufficient that
the following conditions be satisfied:
(i) each of the partial families (u(i)), i E Ij , converges unconditionally;
(ii) the family of sums
(18.2)
S(Ij) = L lu(i)1
iElj
converges unconditionally.
If these conditions are satisfied each of the sums
(18.3)
s(Ij) = L u(i)
iElj
converges unconditionally, the sum of the s(Ij ) converges unconditionally,
and
s(I) = L u(i) = L s(Ij) = L (L U(i)).
iEI
jEJ
jEJ iElj
(1804)
In what follows we shall use the letter s to denote the partial sums of the
family of the u(i) and the letter S to denote those of the family of the lu(i)l.
To prove the necessity of (i), one remarks that the partial sums S(F)
corresponding to all the finite subsets F of I are bounded above, by the definition of unconditional convergence of a sum with positive terms. Similarly
for S(F) for those F contained in a given I j or more generally in any subset
E of Ij but this is precisely the condition for the sum u(i), i E E, to converge
unconditionally (nO 15), whence (i).
To prove that the family of partial sums (2) or (3) is summable, it suffices
to do this for (2) since
Is(Ij)1 :=:; S(Ij)
(nO 15, Theorem 7). Let G be a finite subset of J, and for each element j of
G choose a finite set Fj C Ij and let F be the union, finite, of these Fj . Since
the F j are, like the Ij , pairwise disjoint, we have