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.