2.12. EXERCISES 77

22. Let K be a nonempty closed and convex set in an inner product space (X , |·|) whichis complete. For example, Fn or any other finite dimensional inner product space.Let y /∈ K and let λ = inf{|y− x| : x ∈ K} . Let {xn} be a minimizing sequence. Thatis λ = limn→∞ |y− xn| Explain why such a minimizing sequence exists. Next explainthe following using the parallelogram identity in the above problem as follows.∣∣∣∣y− xn + xm

2

∣∣∣∣2 = ∣∣∣ y2 − xn

2+

y2− xm

2

∣∣∣2=−

∣∣∣ y2− xn

2−( y

2− xm

2

)∣∣∣2 + 12|y− xn|2 +

12|y− xm|2

Hence ∣∣∣∣xm− xn

2

∣∣∣∣2 = −∣∣∣∣y− xn + xm

2

∣∣∣∣2 + 12|y− xn|2 +

12|y− xm|2

≤ −λ2 +

12|y− xn|2 +

12|y− xm|2

Next explain why the right hand side converges to 0 as m,n→ ∞. Thus {xn} is aCauchy sequence and converges to some x ∈ X . Explain why x ∈ K and |x− y|= λ .Thus there exists a closest point in K to y. Next show that there is only one closestpoint. Hint: To do this, suppose there are two x1,x2 and consider x1+x2

2 using theparallelogram law to show that this average works better than either of the two pointswhich is a contradiction unless they are really the same point. This theorem is ofenormous significance.

23. Let K be a closed convex nonempty set in a complete inner product space (H, |·|)(Hilbert space) and let y ∈ H. Denote the closest point to y by Px. Show that Px ischaracterized as being the solution to the following variational inequality

Re(z−Px,y−Px)≤ 0

for all z ∈ K. Hint: Let x ∈ K. Then, due to convexity, a generic thing in K is of theform x+ t (z− x) , t ∈ [0,1] for every z ∈ K. Then

|x+ t (z− x)− y|2 = |x− y|2 + t2 |z− x|2− t2Re(z− x,y− x)

If x = Py, then the minimum value of this on the left occurs when t = 0. Functiondefined on [0,1] has its minimum at t = 0. What does it say about the derivativeof this function at t = 0? Next consider the case that for some x the inequalityRe(z− x,y− x)≤ 0. Explain why this shows x = Py.

24. Using Problem 23 and Problem 22 show the projection map, P onto a closed con-vex subset of a complete inner product space is Lipschitz continuous with Lipschitzconstant 1. That is |Px−Py| ≤ |x− y| .

25. Suppose S is an uncountable set and suppose f (s) is a positive number for each s∈ S.Also let Ŝ denote a finite subset of S. Show that

sup

{∑s∈Ŝ

f (s) : Ŝ⊆ S

}= ∞