Kenneth Kuttler

308 CHAPTER 12. BANACH SPACES

Then with this definition, the fundamental result about orthonormal sets in Hilbert spaceis the following.

Theorem 12.2.13 Let H be a nonempty Hilbert space and let D0 be an orthonor-mal set. Then

1. There exists a maximal orthonormal set D⊇ D0.

2. span(D) = H.

3. If H is separable, then D is countable.

4. In this case where H is separable and D = {dn}∞

n=1 , it follows that for any x∈H,x =∑

∞n=1 (x,dn)dn meaning the series converges in H to x. This is called the Fourier

series of x.

5. Also for x ∈ H, |x|2 = ∑∞n=1 |(x,bn)|2 .

Proof: 1.) First note that there exists such an orthonormal set D0. It could be x ∈ Hwhere |x| = 1. Let F consist of orthonormal sets containing D0 and partially order thesesets by set inclusion. By the Hausdorff maximal theorem, there is a maximal chain C . LetD = ∪C . Then D is orthonormal because C is a chain and any pair of vectors in D must bein a single element of C . D contains D0 and if D is not maximal, there must be some z /∈Dsuch that D∪{z} is also an orthonormal set. But then this contradicts the maximality of Cbecause C ∪{D∪{z}} would be larger chain.

2.) If there exists y ∈H \ span(D), note that span(D) is a closed subspace of H. There-fore, y−Py is not 0 and z ≡ (y−Py)/ |y−Py| satisfies (z,x) = 0 for all x ∈ span(D). Inparticular, (z,x) = 0 for all x∈D. Therefore, D is not maximal after all. Thus span(D) =H.

3.) Now suppose H is separable. If x ̸= y,x,y ∈D, then |x− y|2 = |x|2 + |y|2 = 2 and so|x− y|=

√2. Thus the balls B(x,1) for x ∈ D are disjoint. Since H is separable, there are

only countably many.4.) Let D = {dn}∞

n=1. Let Vk ≡ span{d1, ...,dk} . Now consider the problem of choosingαk to minimize

∣∣x−∑kn=1 αndn

∣∣2 . This expression to minimize equals∣∣∣∣∣x− k

∑n=1

(x,dn)dn +k

∑n=1

((x,dn)−αn)dn

∣∣∣∣∣2

and some algebra using the dn are orthonormal shows this equals∣∣∣∣∣x− k

∑n=1

(x,dn)dn

∣∣∣∣∣2

∑n=1|(x,dn)−αn|2 +2Re

∑n=1

((x,dn)−αn

)(x,dn)

−2Rek

∑n=1

(x,dn)((x,dn)−αn

)Thus the solution to the minimization problem has αn = (x,dn) . Since span(D) = H, itfollows that x = limk→∞ ∑

kn=1 (x,dn)dn

308 CHAPTER 12. BANACH SPACESThen with this definition, the fundamental result about orthonormal sets in Hilbert spaceis the following.Theorem 12.2.13 Let H be a nonempty Hilbert space and let Do be an orthonor-mal set. Then1. There exists a maximal orthonormal set D D Do.2. span(D) =H.3. If H is separable, then D is countable.4. In this case where H is separable and D = {dy},,_, , it follows that for any x € H,x =Ye (x, dn) dn meaning the series converges in H to x. This is called the Fourierseries of x.5. Also for x € H, |x|? =Y2_, |(x,bn)|?.Proof: 1.) First note that there exists such an orthonormal set Do. It could be x € Hwhere |x| = 1. Let ¥ consist of orthonormal sets containing Do and partially order thesesets by set inclusion. By the Hausdorff maximal theorem, there is a maximal chain @. LetD=UG@. Then D is orthonormal because @ is a chain and any pair of vectors in D must bein a single element of @. D contains Do and if D is not maximal, there must be some z ¢ Dsuch that DU {z} is also an orthonormal set. But then this contradicts the maximality of @because @ U{DU {z}} would be larger chain.2.) If there exists y € H \ span (D), note that span (D) is a closed subspace of H. There-fore, y— Py is not 0 and z= (y— Py) /|y—Py| satisfies (z,x) = 0 for all x € span(D). Inparticular, (z,x) =0 for all x € D. Therefore, D is not maximal after all. Thus span (D) = H.3.) Now suppose H is separable. If x 4 y,x,y € D, then |x — y|? = |x|? + |y|? =2 and so|x —y| = V2. Thus the balls B(x, 1) for x € D are disjoint. Since H is separable, there areonly countably many.4.) Let D= {d,}7_,. Let Vi = span {d},...,d,}. Now consider the problem of choosingQ, to minimize |x ~yk, Ondn| . This expression to minimize equals2k kx L (x,dn) dn + d ((x,dy) — On) dnand some algebra using the d, are orthonormal shows this equals2 k k+ ¥ |(t,dy) —om|? +2Re (dn) - On) (x,dn)n=1 n=1kx y (x,dn) dnn=1-2Re Y (sd) (dh)n=1Thus the solution to the minimization problem has @, = (x,d,). Since span(D) = H, itfollows that x = lim... Y*_, (x,dn) dnn=1