28 PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ

pM = hx(x) - h2(x) for x e X,

then the equality of both representations implies that p : X — Y is

compact. But the Frechet derivative of a compact map is compact, so that

1 2

p' (x ) = L (x ) - L (x ) is compact. •

Proposition 2.6 Let f:X — Y be quasi linear Fredholm and be represented

by f(x) = L(x)x+C(x) for xeX. If f:X — Y is Frechet different!able

at xn € X, then

f(xn) - L(x ) is compact.

Proof: Define

pM = f(x) - L(x)(x-xQ) for x € X.

Lemma 2.4 and the differentiability of f : X —-»Y at x imply that

p' (x ) = f (x ) - L(xQ). Also, p : X -^ Y is compact. Thus

f(xn) - L(*n) is compact. D

Lemma 2.7 Let f : X —» Y be quasi linear Fredholm and be represented by

f(x) = L(x)x+ C(x) for x € X.

Let R : X — GL(Y,X) be a parametrix for L. Then

R(x)f(x) = x - 0(x) for x € X,

where \ji : X —* X is compact.

Proof: We have

R(x)L(x) = Id - K(x) for x € X,

where K : X —-»K(X) is continuous. Also,