86 From Classical Mechanics to Quantum Field Theory. A Tutorial
(i)〈x, y〉=〈y,x〉∗;
(ii)〈x, ay+bz〉=a〈x, y〉+b〈x, z〉;
(iii)〈x, x〉≥0 withx=0if〈x, x〉=0.The spaceHis said to be a (complex)Hilbert spaceif it is complete with
respect to the natural norm||x||:=
√
〈x, x〉,x∈H.
In particular, just in view of positivity of the scalar product and regardless the
completeness property, theCauchy-Schwarz inequalityholds
|〈x, y〉|≤||x||||y||,x,y∈H.Another elementary purely algebraic fact is thepolarization identityconcerning
the Hermitian scalar product (here,His not necessarily complete)
4 〈x, y〉=||x+y||^2 −||x−y||^2 −i||x+iy||^2
+i||x−iy||^2 for ofx, y∈H, (2.25)which immediately implies the following elementary result.
Proposition 2.2.1.IfHis a complex vector space with Hermintian scalar product
〈, 〉,alinearmapL:H→Hwhich is an isometry –||Lx||=||x||ifx∈H–
also preserves the scalar product –〈Lx, Ly〉=〈x, y〉forx, y∈H.
The converse proposition is obviously true.
Similar to the above identity, we have another useful identity, for every operator
A:H→H:
4 〈x, Ay〉=〈x+y,A(x+y)〉−〈x−y,A(x−y)〉−i〈x+iy, A(x+iy)〉
+i〈x−iy, A(x−iy)〉 for ofx, y∈H. (2.26)That in particular proves thatif〈x, Ax〉=0for allx∈H,thenA=0.
(Observe that this fact isnot generally true if dealing with real vector spaces
equipped with a real symmetric scalar product.)
We henceforth assume that the reader be familiar with the basic theory of
normed, Banach and Hilbert spaces and notions likeorthogonal of a set,Hilbertian
basis(also calledcomplete orthonormal systems) and that their properties and use
be well known[8; 5]. We summarize here some basic results concerningorthogonal
setsandHilbertian bases.
Notation 2.2.2.IfM⊂H,M⊥:={y∈H|〈y,x〉=0 ∀x∈M}denotes the
orthogonalofM.