after Defn. 1.5.5). By the "converse" of the generalized Radon-
Nikodym Theorem (Theorem 1.5.9), there exist measures
mD: F L(E,Z') and mR: F L(E,Z') (the measures have the same
o
values as the function B since Z' is a dual (cf. part 3(a) of the
statement of this theorem) with finite variation ImD and ImRl
such that:
o o
i) = E(1A<(B. Bz)x,z0>) for A e F, x E E,
z E Z (by taking f = x1A in 1.5.9), and =
0
E(1A<(AR (B ))x,z0>) likewise. Also,
zz
ii) mDJ(A) = E(1IBZ, BzI), and imRI(A) = E(LA l (B )I)
zz
for A c F (we take p = 1A in 1.5.9).
0 0
From (i) we have = E(1A<(B. B )x,z >)
<(mz mZ)(A)x,z0> from earlier. Likewise, =
<(s't s't st + mst)(A)x,z >. Both these hold for all A,
x, z0 so we have mD(A)x = (mz mz)(A)x, and mR(A)x =
(m't ms't st' + mst)(A)x for all A, x; hence mD = m m
S s't' s't st' st
and m = m m mt (and In pa-ticular m mR
have values in L(E,F)).
By (11), we have Imz mZ(A) = |mD (A) = E(1AIBz, B ),
and similarly I't' 't stm + (A) = E(1AIAR (B )I) for
zz
A e F. On the other hand, we have I(mz m)(A)I = Im(Dzz,xA)j [
ml (DzzxA); hence Imz mZJ(A) S JIm(Dzz,xA) = lm]z (A) Im1z(A)
= E(1AVz.) E(1AV ) = E(1A(V z-V )); hence E(1AjBz, Bz)