26 PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ
Proof: Choose a retraction p: A A . According to Theorem 2.1, we may
choose T):A GL(Y,X) a parametrix for a. Now define 0:A —» GL(Y,X)
by
0(A) = 0Q(p(A)) [nCpCX))]"1*^) for A A.
It is clear that 0 is a parametrix for a, and 0(A) = 0 (A) if A A .

Throughout, we will assume that X is compactly embedded in another
Banach space X. Given D £ X, when we refer to its topological properties
we will be referring to the topology induced by X, unless we explicitly
state that we are considering the topology induced by X.
Definition A mapping f : X Y is called quasi linear Fredholm provided
that f has a represent at ion of the form
f(x) = L(x)x + C(x) for x X, (2.3)
where
(i) L : X $n(X,Y) is the restriction to X of a continuous map
L : X $Q(X,Y),
and
(ii) C : X Y is compact.
We will refer to formula (2.3), where (i) and (ii) are satisfied, as a
representat ion of f. We call L : X —- » $ (X,Y) a principal part of f.
If f: X —» Y is any C mapping, we may write f as
f(x) = df(tx)(x)dt + f(0) = L(x)x + f(0) for x e X,
J0
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