388 XV NORMED ALGEBRAS AND SPECTRAL THEORY

and therefore

(1 5.1 0.3.3) junvy . a> T(w) . a = (vw) • u.

The independence asserted above will be a consequence of the following
more precise result :

(1 5.1 0.4) (i) For all x,yeE and all u e ^C(K) we have
(15.10.4.1 ) (T(u) • * | J) = ( u(t) d^y(t).

(ii) If (un) is a uniformly bounded sequence of functions in ^C(K) which
converges to u, then

(15.10.4.2) (T(u) 'X\y)=lim (T(un)

n-+oo

for all x, y in E,

It is enough to prove (15.10.4.2) for x, y lying in a total subspace of E:
the sesquilinear functions (x, y)\->(T(un) -x\y) form an equicontinuous set,
because

boll + ||xo|| * Ib-j^oll + \\x-x0\\ -Ib-

(7.5.5). Take x = Tfa) - a and y = T(s2) ' a, with s1 and s2 in ^C(K); then,
by definition,

(T(un) -x\y) =

a\d) = Is^^ d&

and it is enough to apply the dominated convergence theorem (13.8.4) to the
measure ^. As to (15.10.4.1), it is valid by definition when z/e^c(K), and in
general it follows by applying the dominated convergence theorem twice to
the measure /tx>y, and using (13.7.1).

(15.10.5) For the operator T(u) to be hermitian (resp. positive hermitian
(1 1 .5), resp. zero, resp. unitary) it is necessary and sufficient that u(x) should be
real (resp. u(x) ^ 0, resp. u(x) = 0, resp. \u(x)\ = 1) almost everywhere with
respect to the measure ju.orce up to and including (15.10.7).) The preceding remarks show that the