330 K. IMPLICIT FUNCTION THEOREM
Hence given y EB (0, 8/2) and h EX with 0 < [[h[[ ~ 8/2 - [[y[[, we have
llr1 (y+ h) -r1 (y) - (DJ u-1 (y)))-1 (h)ll
~ 2 llDJ u-l (y)) u-l (y + h) - r^1 (y)) - hll
= o (llr^1 (y + h) - 1-^1 (y)ll)= 0 ([[h[[))
where we also used (K.5). Therefore, 1-^1 is differentiable in B (0, 8 /2). Since D f
and 1-^1 are continuous, we have that
D u-1) (y) =(DJ u-1 (y))rl
is a continuous function of y; i.e., 1-^1 is C^1. DA simple consequence of the Banach space inverse function theorem is the
following Banach space implicit function theorem.
THEOREM K.4 (Banach space IFT). Let X, Y , and Z be Banach spaces, let
Uc Xx y be an open set, and let f : U -t Z be a Ck map, where k;:::: l. Suppose
(xo, Yo) EU is such that the partial derivative (oyf)(xo,yo) : Y -t Z, defined by
(o !) (v) = lim f (xo, Yo+ sv) - f (xo, Yo)
Y (xo,Yo) s--tO Sfor v E Y, is a Banach space isomorphism. Then there exists c > 0 and 8 > 0 such
that for every x EB (x 0 , c) there exists a unique solution y ~ </> (x) EB (yo, 8) to
the equation
f (x, y) = f (xo, Yo).
The map</>: B (x 0 , c) -t B (y 0 , 8) is Ck and its derivative D</>: X -t Y is given by(D)x = - ((oyf)cx,(x)))-
1(^0) (oxf)cx,
where ox f : X -t Z is the partial derivative.
1.4. Banach manifolds.
Just as a manifold is a topological space modeled on a Euclidean space (of a
fixed dimension), a Banach manifold is a topological space modeled on (possibly
different) Banach spaces.
A Ck Banach manifold, where k E N U { 0, oo}, is defined to be a set B with
a Ck Banach manifold atlas for B. Such an atlas is defined to be a collection of
subsets {Ui} iEJ which cover B, together with bijections
<fJi : ui --+ <fJi (Ui) c xi,
where
(1) Xi is a Banach space and <fJi (Ui) is an open subset,
(2) <fJi (Ui n Uj) c Xi is open, and
(3) the overlap map
<{Jj 0 <pi^1 : <fJi (Ui n Uj) -t <{Jj (Ui n Uj)
n nis Ck for all i,j EI.