Real and Complex Analysis (Rudin)

78 REAL AND COMPLEX ANALYSIS
(b) If J1. is any positive measure, L2(J1.) is a Hilbert space, with inner product
(f, g) = Lfg dJ1..
The integrand on the right is in L I (J1.), by Theorem 3.8, so that (f, g) is
well defined. Note that
Ilfll = (f,f) 1 /2 = {Llfl 2 dJ1.f/2 = IIf1l2'
The completeness of I3(J1.) (Theorem 3.11) shows that I3(J1.) is indeed a
Hilbert space. [We recall that I3(J1.) should be regarded as a space of
equivalence classes of functions; compare the discussion in Sec. 3.10.]
For H = I3(J1.), the inequalities 4.2 and 4.3 turn out to be special
cases of the inequalities of HOlder and Minkowski.
Note that Example (a) is a special case of (b). What is the measure in
(a)?
(c) The vector space of all continuous complex functions on [0, 1] is an
inner product space if
r (f, g) = f(t)g(t) dt
but is not a Hilbert space.
4.6 Theorem For any fixed y E H, the mappings
are continuous functions on H.
x--+ (x, y), x--+ (y, x), x--+ IIxll
PROOF The Schwarz inequality implies that
I (Xl' y) - (X2' y) I = I (Xl - X 2 , y) I ::s;; IIXI - x211 Ilyll,
which proves that x--+ (x, y) is, in fact, uniformly continuous, and the same is
true for x--+ (y, x). The triangle inequality IIxIl1 ::s;; Ilxl - x211 + IIx21! yields
IlxI11 - IIx211 ::S;;-lIxl - x211,
and if we interchange Xl and X2 we see that
Illxlll - IIx2111 ::s;; Ilxl - x211
for all Xl and X2 E H. thus x--+ Ilxll is also uniformly continuous.
IIII
4.7 Subspaces A subset M of a vector space V is called a subspace of V if M is
itself a vector space, relative to the addition and scalar multiplication which are
defined in V. A necessary and sufficient condition for a set MeV to be a subspace
is that X + Y E M and IXX E M whenever X and y E M and IX is a scalar.