Appendix to Chapter III
315
Example 1. Consider a set X (in practice, finite or countable) and the set
L2(X) of functions f(x) defined on X, with complex values, and such that
(5.3)
L If(x)1 2 < +00
where of course we are dealing with unconditional convergence; compare with
the space L1(Z) of Chap. II, nO 18, example 3. If f, g E L2(X), then, for every
finite subset F of X,
(
)
1/2
(
) 1/2 (
) 1/2
(5.4)
~ If(x) + g(x)1 2
::; ~ If(x)1 2
+ ~ Ig(x)12
by the Cauchy-Schwarz inequality for a finite sum. Passing to the limit over F,
one deduces that the series E If(x) + g(x)12, extended over all X, converges,
so that f + g E L2(X). Since the product of an f E L2(X) by a constant is
clearly in L2(X), one concludes that L2(X), endowed with usual operations
on scalar functions, is a complex vector space. Further, the series
(5.5)
(f I g) = L f(x)g(x)
converges absolutely or unconditionally for all f, g E L2(X), as seen by applying Cauchy-Schwarz to its finite partial sums. This scalar product clearly
satisfies the preceding conditions, including (H 4). The corresponding norm
is denoted by
(5.6)
( ""
2) 1/2
IIfl12 = L..; If(x)1
.
The formulae are the same as in CP: the values f(x) play the role of "coordinates" of the ''vector'' f. But the vector space L2(X) is of infinite dimension
if X is infinite.
The space L2(X) is complete. Consider a Cauchy sequence (fn) in this
space. Since obviously Ifp(x) - fq(x)1 ::; Ilfp - fql12 = d (fp,fq) for any
x E X, the given functions fn(x) converge simply to a limit function f(x)
by Cauchy's criterion in C, and clearly If(x)1 2 = lim Ifn(x)1 2 for all x E X.
For every finite subset F of X we then have
"" If(x)1 2 = lim "" Ifn(x)1 2 ::; lim "" Ifn(x)1 2 = lim Ilfnl1 2 ;
L.,.;
n--i'OO ~
n--+(X) L-t
n-+oo
xEF
xEF
xEX
the last limit exists because I Ilfpll-llfqll I ::; Ilfp - fqll in any normed vector
space.
The series E If(x)1 2 is therefore convergent, whence f E L2(X) and
IIfll2 ::; lim Ilfnl12· In this inequality we replace the sequence (fn) by the
sequence (fn - fp), for a given p: it also satisfies Cauchy's criterion, and
converges simply to f - f p , so we obtain Ilf - fpl12 ::; limn Ilfn - fp112;