From Classical Mechanics to Quantum Field Theory

88 From Classical Mechanics to Quantum Field Theory. A Tutorial

Finally, ifN⊂His an Hilbertian basis, the decompositions hold for every
x, y∈H

x=

∑

u∈N

〈u, x〉u, 〈x, y〉=

∑

u∈N

〈x, u〉〈u, y〉.

In view of the already mentioned fact that only a finite or countable set of el-
ements〈un,x〉do not vanish, the first sum is actually a finite sum or at most
series limm→+∞

∑m
n=0〈un,x〉uncomputed with respect to the norm ofH,where
the order used to label the elementundoes not matter as the series absolutely
converges. The second sum is similarly absolutely convergent so that, again, it
can be re-ordered arbitrarily.
We remind the reader of the validity of an elementary though fundamental
tecnical result (e.g., see[8; 5]):

Theorem 2.2.6(Riesz’ lemma).LetHbe a complex Hilbert space. φ:H→C
is linear and continuous if and only if it has the formφ=〈x, 〉for somex∈H.
The vectorxis uniquely determined byφ.

2.2.2 Typesofoperators......................

Our goal is to present some basic results ofspectral analysis,usefulinQM.
From now on, anoperatorAinHalways means alinearmapA:D(A)→H,
whosedomain,D(A)⊂H,isasubspaceofH.Inparticular,Ialways denotes
theidentity operatordefined on thewholespace (D(I)=H)

I:Hx→x∈H.

IfAis an operator inH,Ran(A):={Ax|x∈D(A)}is therange(also known as
image)ofA.

Definition 2.2.7. IfAandBare operators inH,

A⊂Bmeans thatD(A)⊂D(B)andB|D(A)=A,

where|Sis the standard “restriction toS” symbol. We also adopt usual conven-
tions regardingstandard domainsof combinations of operatorsA, B:

(i)D(AB):={x∈D(B)|Bx∈D(A)}is the domain ofAB;

(ii)D(A+B):=D(A)∩D(B)is the domain ofA+B;

(iii)D(αA)=D(A)forα=0is the domain ofαA.

With these definitons, it is easy to prove that

(1) (A+B)+C=A+(B+C);

(2)A(BC)=(AB)C;