202 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Lemma 8.7.1 Let g : U → Rp be C2 where U is an open subset of Rp. Then

p

∑j=1

cof(Dg)i j, j = 0,

where here (Dg)i j ≡ gi, j ≡ ∂gi∂x j

. Also, cof(Dg)i j =∂ det(Dg)

∂gi, j.

Definition 8.7.2 Let h be a function defined on an open set, U ⊆ Rp. Then h ∈Ck(U)

if there exists a function g defined on an open set, W containng U such that g = hon U and g is Ck (W ) .

Lemma 8.7.3 There does not exist h ∈ C2(

B(0,R))

such that h : B(0,R)→ ∂B(0,R)which also has the property that h(x) = x for all x ∈ ∂B(0,R) ≡ {x : |x|= R} Such afunction is called a retract.

Proof: If h is such a retract, then for all x ∈ B(0,R) ,det(Dh(x)) = 0. This is becauseif det(Dh(x)) ̸= 0 for some such x, then by the inverse function theorem, h(B(x,δ )) is anopen set for small enough δ but this would require that this open set is a subset of ∂B(0,R)which is impossible because no open ball is contained in ∂B(0,R). Here and below, let BRdenote B(0,R).

Now suppose such an h exists. Let λ ∈ [0,1] and let

pλ (x)≡ x+λ (h(x)−x) .

This function, pλ is a homotopy of the identity map and the retract h. Let

I (λ )≡∫

B(0,R)det(Dpλ (x))dx.

Then using the dominated convergence theorem,

I′ (λ ) =∫

B(0,R)∑i. j

∂ det(Dpλ (x))∂ pλ i, j

∂ pλ i j (x)∂λ

dx

=∫

B(0,R)∑

i∑

j

∂ det(Dpλ (x))∂ pλ i, j

(hi (x)− xi), j dx

=∫

B(0,R)∑

i∑

jcof(Dpλ (x))i j (hi (x)− xi), j dx

Now by assumption, hi (x)= xi on ∂B(0,R) and so one can integrate by parts, in the iteratedintegrals used to compute

∫B(0,R) and write

I′ (λ ) =−∑i

∫B(0,R)

∑j

cof(Dpλ (x))i j, j (hi (x)− xi)dx = 0.

Therefore, I (λ ) equals a constant. However, I (0) = mp (B(0,R)) ̸= 0 and as pointed outabove, I (1) = 0. ■

The following is the Brouwer fixed point theorem for C2 maps.