422 CHAPTER 17. INNER PRODUCT SPACES

22. Show that in any inner product space the parallelogram identity holds.

|x+y|2 + |x−y|2 = 2 |x|2 +2 |y|2

Next show that in a real inner product space, the polarization identity holds.

⟨x,y⟩= 14

(|x+y|2−|x−y|2

).

23. ∗This problem is for those who know about Cauchy sequences and completeness ofFp and about closed sets. Suppose K is a closed nonempty convex subset of a finitedimensional subspace U of an inner product space V . Let y ∈ V. Then show thereexists a unique point x ∈ K which is closest to y. Hint: Let

λ = inf{|y− z| : z ∈ K}

Let {xn} be a minimizing sequence,

|y−xn| → λ .

Use the parallelogram identity in the above problem to show that {xn} is a Cauchysequence. Now let {uk}p

k=1 be an orthonormal basis for U . Say

xn =p

∑k=1

cnkuk

Verify that for cn ≡(cn

1, · · · ,cnp)∈ Fp

|xn−xm|= |cn− cm|Fp .

Now use completeness of Fp and the assumption that K is closed to get the existenceof x ∈ K such that |x−y|= λ .

24. ∗Let K be a closed nonempty convex subset of a finite dimensional subspace U ofa real inner product space V . (It is true for complex ones also.) For x ∈ V, denoteby Px the unique closest point to x in K. Verify that P is Lipschitz continuous withLipschitz constant 1,

|Px−Py| ≤ |x−y| .Hint: Use Problem 21.

25. ∗ This problem is for people who know about compactness. It is an analysis problem.If you have only had the usual undergraduate calculus course, don’t waste your timewith this problem. Suppose V is a finite dimensional normed linear space. Recallthis means that there exists a norm ∥·∥ defined on V as described above,

∥v∥ ≥ 0 equals 0 if and only if v = 0

∥v+u∥ ≤ ∥u∥+∥v∥ , ∥αv∥= |α|∥v∥ .Let |·| denote the norm which comes from Example 17.1.3, the inner product bydecree. Show |·| and ∥·∥ are equivalent. That is, there exist constants δ ,∆ > 0 suchthat for all x ∈V,

δ |x| ≤ ∥x∥ ≤ ∆ |x| .Explain why every two norms on a finite dimensional vector space must be equivalentin the above sense.