From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 111

(2.51) implies thatHx→

∫

Xf(λ)dμ

(xyP)(λ) is continuous atx=0. This

map is also anti-linear as follows from the definition ofμx,y.Anelementaryuse
of Riesz’ lemma proves that there exists a vector, indicated by

∫

Xf(λ)dP(λ)y,
satisfying (2.48). That is the action of an operator on a vectory∈Δfbecause
Δfy→

∫

Xf(λ)dμ

(P)

xy(λ) is linear. 

Remark 2.2.51. Identity (2.50) givesΔfa direct meaning in terms of bound-
edness of

∫

Xf(λ)dP(λ).Sinceμxx(X)=||x||

(^2) <+∞, (2.50) together with the
definition ofΔf immediately implies that: iff is bounded or, more weaklyP-
essentially bounded^8 onX,then
∫
X
f(λ)dP(λ)∈B(H)
and
∣∣
∣∣

∣∣

∣∣

∫

X

f(λ)dP(λ)

∣∣

∣∣

∣∣

∣∣≤||f||(∞P)≤||f||∞.

TheP-essentially boundedness is also anecessary(not only sufficient) condition
for

∫

Xf(λ)dP(λ)∈B(H)[8; 5; 6].

Exercise 2.2.52.
(1)Prove inequality (2.51).

Solution. Letx∈Handy∈Δf.Ifs:X→Cis asimple functionand
h:X→Cis theRadon-Nikodym derivativeofμxywith respect to|μxy|so that
|h(x)|=1andμxy(E)=

∫

Ehd|μxy|(see, e.g.,[^5 ]), we have for an increasing
sequence of simple functionszn→hpointwise, with|zn|≤|h−^1 |= 1, due to the
dominate convergence theorem,

∫

X

|s|d|μxy|=

∫

X

|s|h−^1 dμxy= limn→+∞

∫

X

|s|zndμxy= limn→+∞

〈

x,

∑Nn

k=1

zn,kPEn,ky

〉

.

In the last step, we have made use of (iii)(b) in remark 2.2.48 for the simple
function|s|zn=

∑Nn

k=1zn,kχEn,k. Cauchy Schwartz inequality immediately yields

∫

X

|s|d|μxy|≤||x||n→lim+∞

∣∣

∣∣

∣

∣∣

∣∣

∣

∑Nn

k=1

zn,kPEn,ky

∣∣

∣∣

∣

∣∣

∣∣

∣=||x||n→lim+∞

√∫

X

|szn|^2 dμyy,

(^8) As usual,||f||∞(P)is the infimum of positive realsrsuch thatP({x∈X||f(x)|>r})=0.