576 CHAPTER 22. HILBERT SPACES

Lemma 22.1.4 In an inner product space,

∥x+ y∥2 +∥x− y∥2 = 2∥x∥2 +2∥y∥2.

The proof, a straightforward application of the inner product axioms, is left to thereader.

Lemma 22.1.5 For x ∈ H, an inner product space,

∥x∥= sup∥y∥≤1

|(x,y)| (22.4)

Proof: By the Cauchy Schwarz inequality, if x ̸= 0,

∥x∥ ≥ sup∥y∥≤1

|(x,y)| ≥(

x,x∥x∥

)= ∥x∥ .

It is obvious that 22.4 holds in the case that x = 0.

Definition 22.1.6 A Hilbert space is an inner product space which is complete.Thus a Hilbert space is a Banach space in which the norm comes from an inner product asdescribed above.

In Hilbert space, one can define a projection map onto closed convex nonempty sets.

Definition 22.1.7 A set K is convex if whenever λ ∈ [0,1] and x,y ∈ K, λx+(1−λ )y ∈ K.

Theorem 22.1.8 Let K be a closed convex nonempty subset of a Hilbert space H,and let x ∈H. Then there exists a unique point Px ∈ K such that ∥Px− x∥ ≤ ∥y− x∥ for ally ∈ K.

Proof: Consider uniqueness. Suppose that z1 and z2 are two different elements of Ksuch that for i = 1,2,

∥zi− x∥ ≤ ∥y− x∥ (22.5)

for all y ∈ K. Also, note that since K is convex, z1+z22 ∈ K. Therefore, by the parallelogram

identity,

∥z1− x∥2 ≤∥∥∥∥ z1 + z2

2− x∥∥∥∥2

=

∥∥∥∥ z1− x2

+z2− x

2

∥∥∥∥2

= 2(∥∥∥∥ z1− x

2

∥∥∥∥2

+

∥∥∥∥ z2− x2

∥∥∥∥2

)−∥∥∥∥ z1− z2

2

∥∥∥∥2

=12∥z1− x∥2 +

12∥z2− x∥2−

∥∥∥∥ z1− z2

2

∥∥∥∥2

≤ ∥z1− x∥2−∥∥∥∥ z1− z2

2

∥∥∥∥2

,

where the last inequality holds because of 22.5. Hence z1 = z2 after all and this showsuniqueness.