790 CHAPTER 23. DEGREE THEORY

Proof: Letting {v1, · · · ,vm} be a basis for Vm, let a basis for V be

{v1, · · · ,vm,vm+1, · · · ,vn}

Let θ be the isomorphism which satisfies θei = vi where the ei denotes the standard basisvectors for Rn. Then from the above,

d (I− f ,Ω,y) ≡ d(θ−1 ◦ (I− f )◦θ ,θ−1

Ω,θ−1y)

= d(

θ−1 ◦ (I− f )◦θ |

θ−1(Ω)∩Rn

m,θ−1 (Ω)∩Rn

m,θ−1y)

≡ d((I− f ) |

Ω∩Vm,Ω∩Vm,y

).

23.9 The Leray Schauder DegreeThis is a very important generalization to Banach spaces. It turns out you can define thedegree of I−F where F is a compact mapping. To recall what one of these is, here is thedefinition.

Definition 23.9.1 Let Ω be a bounded open set in X a Banach space and let F : Ω→ X becontinuous. Then F is called compact if F (B) is precompact whenever B is bounded. Thatis, if {xn} is a bounded sequence, then there is a subsequence

{xnk

}such that

{F(xnk

)}converges.

Theorem 23.9.2 Let F : Ω→ X as above be compact. Then for each ε > 0, there existsFε : Ω→ X such that F has values in a finite dimensional subspace of X and

supx∈Ω

∥Fε (x)−F (x)∥< ε

In addition to this, (I−F)−1 (compact set) =compact set. (This is called “proper”.)

Proof: It is known that F (Ω) is compact. Therefore, there is an ε net for F (Ω),{Fxk}n

k=1 satisfyingF (Ω)⊆ ∪kB(Fxk,ε)

Now letφ k (Fx)≡ (ε−∥Fx−Fxk∥)+

Thus this is equal to 0 if ∥Fxk−Fx∥ ≥ ε and is positive if ∥Fxk−Fx∥< ε . Then consider

Fε (x)≡n

∑k=1

F (xk)φ k (Fx)

∑i φ i (Fx)

It clearly has values in span({Fxk}nk=1) . How close is it to F (x)? Say Fx ∈ B(Fxk,ε) .

Then for such x,∥F (x)−F (xk)∥< ε by definition. Hence

∥F (x)−Fε (x)∥ = ∑k:∥F(x)−Fxk∥<ε

∥F (xk)−F (x)∥ φ k (Fx)∑i φ i (Fx)

< ε ∑k

φ k (Fx)∑i φ i (Fx)

= ε