§2. Absolutely convergent series
147
The simplest example of a noncommutative group is obtained by considering a set X and the set 6(X) of bijective maps X ~ X, i.e. of permutations of X, the "product" of two such permutations being the composed map
(Chap. I)j we know that if X is finite with n elements then 6(X) possesses n!
elements. Linear algebra provides other examples of noncommutative groups:
the group GLn(K) of invertible matrices n x n with coefficients in a field K,
for example Q, lR or C, or even with coefficients in a ring such as Z, for
example the sets of matrices
(: !) with a,b,e,d E Z, ad - be = ±1,
the "orthogonal" matrices with coefficients in K, the "unitary" matrices with
coefficients in C, etc. There are countless interesting examples.
That said, consider an arbitrary group G (not endowed with a ''topology" or, as one says, "discrete") and denote by Ll(G) the set of functions
J : G ~ C such that
(18.25)
IIJlll = L IJ(x)1 < +00.
xEG
For two such functions, put
(18.26) J * g(x) = L J(y)g(z) = L J(xz-1)g(z) = L J(y)g(y-lx)
yz=x
zEG
yEG
as in (21). Everything we have said about the convolution product in Z extends to this case - except the formula J * 9 = 9 * J if G is not commutative
- and with exactly the same proofs, since, for a E G given, the maps x 1--4 ax
and x 1--4 xa are permutations of the set G. For example, to prove the associativity of the convolution product, one starts from the series
L J(x)g(y)h(z)
xyz=u
extended over all the systems of elements of G such that xy z has a given
value U E Gj it converges unconditionally since it is a partial sum of a product
of three absolutely convergent seriesj one can therefore group the terms ad
libitum. If you group them according to the value of the product yz you find
the value at the point u E G of J * (g * h)j if you group them according to
the value of the product xy you find the value at u of (f * g) * h, qed.
Moral: unconditional convergence allows one to apply the rules oj algebra
valid Jor finite sums to infinite sums also. There are no problems of convergence at all, since one knows in advance that everything converges.
147
The simplest example of a noncommutative group is obtained by considering a set X and the set 6(X) of bijective maps X ~ X, i.e. of permutations of X, the "product" of two such permutations being the composed map
(Chap. I)j we know that if X is finite with n elements then 6(X) possesses n!
elements. Linear algebra provides other examples of noncommutative groups:
the group GLn(K) of invertible matrices n x n with coefficients in a field K,
for example Q, lR or C, or even with coefficients in a ring such as Z, for
example the sets of matrices
(: !) with a,b,e,d E Z, ad - be = ±1,
the "orthogonal" matrices with coefficients in K, the "unitary" matrices with
coefficients in C, etc. There are countless interesting examples.
That said, consider an arbitrary group G (not endowed with a ''topology" or, as one says, "discrete") and denote by Ll(G) the set of functions
J : G ~ C such that
(18.25)
IIJlll = L IJ(x)1 < +00.
xEG
For two such functions, put
(18.26) J * g(x) = L J(y)g(z) = L J(xz-1)g(z) = L J(y)g(y-lx)
yz=x
zEG
yEG
as in (21). Everything we have said about the convolution product in Z extends to this case - except the formula J * 9 = 9 * J if G is not commutative
- and with exactly the same proofs, since, for a E G given, the maps x 1--4 ax
and x 1--4 xa are permutations of the set G. For example, to prove the associativity of the convolution product, one starts from the series
L J(x)g(y)h(z)
xyz=u
extended over all the systems of elements of G such that xy z has a given
value U E Gj it converges unconditionally since it is a partial sum of a product
of three absolutely convergent seriesj one can therefore group the terms ad
libitum. If you group them according to the value of the product yz you find
the value at the point u E G of J * (g * h)j if you group them according to
the value of the product xy you find the value at u of (f * g) * h, qed.
Moral: unconditional convergence allows one to apply the rules oj algebra
valid Jor finite sums to infinite sums also. There are no problems of convergence at all, since one knows in advance that everything converges.