558 CHAPTER 19. HILBERT SPACES

Thus A∗y∗nk(x)→ A∗y∗ (x) for every x. In addition to this, for x ∈ B,∣∣∣∣A∗y∗ (x)−A∗y∗nk

(x)∣∣∣∣ =

∣∣∣∣y∗ (Ax)− y∗nk(Ax)

∣∣∣∣=

∣∣∣∣g(Ax)− y∗nk(Ax)

∣∣∣∣=

∣∣∣∣∣∣∣∣||x|| f (A(

x||x||

))−||x||y∗nk

(Ax||x||

)∣∣∣∣∣∣∣∣≤

∣∣∣∣∣∣∣∣ f (A(

x||x||

))− y∗nk

(Ax||x||

)∣∣∣∣∣∣∣∣and this is uniformly small for large k due to the uniform convergence of y∗nk

to f on A(B).Therefore,

∣∣∣∣A∗y∗−A∗y∗nk

∣∣∣∣→ 0.

19.10 The Fredholm AlternativeRecall that if A is an n× n matrix and if the only solution to the system, Ax = 0 is x = 0then for any y ∈ Rn it follows that there exists a unique solution to the system Ax = y.This holds because the first condition implies A is one to one and therefore, A−1 exists. Ofcourse things are much harder in a general Banach space. Here is a simple example for aHilbert space.

Example 19.10.1 Let L2 (N; µ) = H where µ is counting measure. Thus an element of His a sequence, a = {ai}∞

i=1 having the property that

||a||H ≡

(∞

∑k=1|ak|2

)1/2

< ∞.

Define A : H→ H byAa≡ b≡{0,a1,a2, · · ·} .

Thus A slides the sequence to the right and puts a zero in the first slot. Clearly A is one toone and linear but it cannot be onto because it fails to yield e1 ≡ {1,0,0, · · ·}.

Notwithstanding the above example, there are theorems which are like the linear alge-bra theorem mentioned above which hold in an arbitrary Banach spaces in the case wherethe operator is compact. To begin with here is an interesting lemma.

Lemma 19.10.2 Suppose A ∈L (X ,X) is compact for X a Banach space. Then it followsthat (I−A)(X) is a closed subspace of X.

Proof: Suppose (I−A)xn→ y. Let

αn ≡ dist(xn,ker(I−A))

and let zn ∈ ker(I−A) be such that

αn ≤ ||xn− zn|| ≤(

1+1n

)αn.