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,

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