From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 87

EvidentlyM⊥is a closed subspace ofH. ⊥enjoys several nice properties quite
easy to prove (e.g., see[8; 5]), in particular,

span(M)=(M⊥)⊥ and H=span(M)⊕M⊥ (2.27)

where the bar denotes the topological closure and⊕the direct orthogonal sum.

Definition 2.2.3.A Hilbertian basisNof a Hilbert spaceHis a set oforthonor-
mal vectors, i.e.,||u||=1and〈u, v〉=0foru, v∈Nwithu=v, such that if
s∈Hsatisfies〈s, u〉=0for everyu∈Hthens=0.

Hilbertian bases always exist as a consequence of Zorn’s lemma. Notice that, as
consequence of (2.27), the following proposition is valid

Proposition 2.2.4. N⊂His a Hilbertian basis of the Hilbert spaceHif and
only ifspan(N)=H.

Furthermore, a generalized version ofPythagorean theoremholds true.

Proposition 2.2.5. N⊂His a Hilbertian basis of the Hilbert spaceHif and
only if

||x||^2 =

∑

u∈N

|〈u, x〉|^2 for everyx∈H.

The sum appearing in the above decomposition ofxis understood as thesupremum
of the sums

∑

u∈F|〈u, x〉|

(^2) for every finite setF⊂N. As a consequence (e.g., see
[8; 5]), only a set at most countable of elements|〈u, x〉|^2 do not vanish and the
sum is therefore interpreted as a standard series that can be re-ordered preserving
the sum because it absolutely converges. It turns out that all Hilbertian bases
ofHhave the same cardinality andHis separable (it admits a dense counteble
subset) if and only ifHhas an either finite our countable Hilbertian basis. If
M⊂His a set of orthonormal vectors, the waeker so-calledBessel inequality
holds
||x||^2 ≥

∑

u∈M

|〈u, x〉|^2 for everyx∈H,

so that Hilbertian bases are exactly orthonormal sets saturating the inequality.