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,
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