578 CHAPTER 22. HILBERT SPACES

If and only if for all t ∈ [0,1],

t2 ∥y− z∥2−2t Re(x− z,y− z)≥ 0. (22.7)

Now this is equivalent to 22.7 holding for all t ∈ (0,1). Therefore, dividing by t ∈ (0,1) ,22.7 is equivalent to

t ∥y− z∥2−2Re(x− z,y− z)≥ 0

for all t ∈ (0,1) which is equivalent to 22.6. ■

Corollary 22.1.10 Let K be a nonempty convex closed subset of a Hilbert space, H.Then the projection map P is continuous. In fact,|Px−Py| ≤ |x− y| .

Proof: Let x,x′ ∈ H. Then by Corollary 22.1.9,

Re(x′−Px′,Px−Px′

)≤ 0, Re

(x−Px,Px′−Px

)≤ 0

Hence

0 ≤ Re(x−Px,Px−Px′

)−Re

(x′−Px′,Px−Px′

)= Re

(x− x′,Px−Px′

)−∣∣Px−Px′

∣∣2and so |Px−Px′|2 ≤ |x− x′| |Px−Px′| . ■

The next corollary is a more general form for the Brouwer fixed point theorem.

Corollary 22.1.11 Let f : K→ K where K is a convex compact subset of Rn. Then fhas a fixed point.

Proof: Let K ⊆ B(0,R) and let P be the projection map onto K. Then consider themap f ◦P which maps B(0,R) to B(0,R) and is continuous. By the Brouwer fixed pointtheorem for balls, this map has a fixed point. Thus there exists x such that (f ◦P)(x) = x.Now the equation also requires x ∈ K and so P(x) = x. Hence f (x) = x. ■

Recall the following definition from linear algebra about direct sum notation.

Definition 22.1.12 Let H be a vector space and let U and V be subspaces. U ⊕V = H if every element of H can be written as a sum of an element of U and an element ofV in a unique way.

The case where the closed convex set is a closed subspace is of special importance andin this case the above corollary implies the following.

Corollary 22.1.13 Let K be a closed subspace of a Hilbert space H, and let x ∈ H.Then for z ∈ K, z = Px if and only if

(x− z,y) = 0 (22.8)

for all y ∈ K. Furthermore, H = K⊕K⊥ where

K⊥ ≡ {x ∈ H : (x,k) = 0 for all k ∈ K}

and∥x∥2 = ∥x−Px∥2 +∥Px∥2 . (22.9)

578 CHAPTER 22. HILBERT SPACESIf and only if for all ¢ € [0, 1],? |ly—z||° —2rRe(x-z,y—z) 50. (22.7)Now this is equivalent to 22.7 holding for all t € (0,1). Therefore, dividing by ¢ € (0,1),22.7 is equivalent tot||y 2] -2Re(x—z,y—z) > 0for all t € (0,1) which is equivalent to 22.6.Corollary 22.1.10 Let K be a nonempty convex closed subset of a Hilbert space, H.Then the projection map P is continuous. In fact,|Px — Py| < |x—y|.Proof: Let x,x’ € H. Then by Corollary 22.1.9,Re (x — Px',Px— Px’) <0, Re (x — Px, Px’ — Px) <0Hencei)lARe (x — Px, Px — Px’) —Re (x! — Px',Px— Px’)= Re (x —x', Px — Px’) — |Px— Px! |?and so |Px — Px'|? < |x —2x/||Px— Px’|.The next corollary is a more general form for the Brouwer fixed point theorem.Corollary 22.1.11 Let f : K — K where K is a convex compact subset of R". Then fhas a fixed point.Proof: Let K C B(0,R) and let P be the projection map onto K. Then consider themap f oP which maps B(0,R) to B(0,R) and is continuous. By the Brouwer fixed pointtheorem for balls, this map has a fixed point. Thus there exists x such that (f oP) (a) =a.Now the equation also requires « € K and so P(x) = a. Hence f (x)= a.Recall the following definition from linear algebra about direct sum notation.Definition 22.1.12 Let H be a vector space and let U and V be subspaces. U®V =H if every element of H can be written as a sum of an element of U and an element ofV ina unique way.The case where the closed convex set is a closed subspace is of special importance andin this case the above corollary implies the following.Corollary 22.1.13 Let K be a closed subspace of a Hilbert space H, and let x € H.Then for z © K, z= Px if and only if(x—z,y) =0 (22.8)for all y € K. Furthermore, H = K @ K+ whereKt = {x EH: (x,k) =O forall k € K}and\|x||? = |lx— Px||? + ||Px|]?. (22.9)