Chapter 22

Hilbert SpacesIn this chapter, Hilbert spaces, which have been alluded to earlier are given a completediscussion. These spaces, as noted earlier are just complete inner product spaces.

22.1 Basic TheoryDefinition 22.1.1 Let X be a vector space. An inner product is a mapping fromX × X to C if X is complex and from X × X to R if X is real, denoted by (x,y) whichsatisfies the following.

(x,x)≥ 0, (x,x) = 0 if and only if x = 0, (22.1)

(x,y) = (y,x). (22.2)

For a,b ∈ C and x,y,z ∈ X,

(ax+by,z) = a(x,z)+b(y,z). (22.3)

Note that 22.2 and 22.3 imply (x,ay+bz) = a(x,y)+b(x,z). Such a vector space is calledan inner product space.

The Cauchy Schwarz inequality is fundamental for the study of inner product spaces.

Theorem 22.1.2 (Cauchy Schwarz) In any inner product space

|(x,y)| ≤ ∥x∥ ∥y∥ .

Equality holds if and only if x is a multiple of y. The inequality holds under the weakerassumption that (x,x)≥ 0 without the stipulation that this happens only if x = 0.

Proof: Let ω ∈C, |ω|= 1, and ω(x,y) = |(x,y)|= Re(x,yω). Let F(t) = (x+tyω,x+tωy), t ∈R. If y= 0 there is nothing to prove because (x,0) = (x,0+0) = (x,0)+(x,0) andso (x,0) = 0. Thus, it can be assumed y ̸= 0. Then from the axioms of the inner product,

F(t) = ∥x∥2 +2t Re(x,ωy)+ t2 ∥y∥2 ≥ 0.

This yields ∥x∥2+2t|(x,y)|+t2 ∥y∥2≥ 0. Since this inequality holds for all t ∈R, it followsfrom the quadratic formula that 4|(x,y)|2−4∥x∥2 ∥y∥2 ≤ 0. In getting this inequality, it wasonly necessary to assume (x,x)≥ 0.

Consider the claim about equality. Note that if x = αy, then equality holds directly.Indeed this is the only way this can happen because if equality holds in the Cauchy Schwarzinequality, F (t) = 0 for some real t. This happens only if x =−tωy for some t real. ■

Proposition 22.1.3 For an inner product space, ∥x∥ ≡ (x,x)1/2 does specify a norm.

Proof: All the axioms are obvious except the triangle inequality. To verify this,

∥x+ y∥2 ≡ (x+ y,x+ y)≡ ∥x∥2 +∥y∥2 +2Re(x,y)

≤ ∥x∥2 +∥y∥2 +2 |(x,y)|≤ ∥x∥2 +∥y∥2 +2∥x∥∥y∥= (∥x∥+∥y∥)2. ■

The following lemma is called the parallelogram identity.

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