8 WILHELM KLINGENBERG

is a fibre map of some 6' X fi' C ß X (5, È' open, into the bündle

L(TT; i):L(£;F)-*/.

Consider the associated continous mapping r:

Hl(6'

X 0') -»

H}(L(E;

F)).

Then, with T2f: //'(è') - H\L{E\ F)\

const||r(£, 11)11,1111 — €11,.

Since Ñ(î,ae) = 0 we get ||f(£, ç)||, - 0 with ÉÉ£-ç||,-0 , i.e., r

2

f G

H\L(H\E)\ H\F))) has the properties of a differential, ?/ = Ã2/~. •

There exists an open neighborhood 0 of the 0-section of ô: ÃÌ - Ì with the

property that exp \(È ñ= ÔñÌ Ð È) is a, diffeomorphism onto its image = open

neighborhood of ñ GM. For Ì compact, we can choose 0 to be the å-ball bündle

of ô, some small å 0, i.e., 0^ = Be(0p), all ñ G Ì.

1.10

DEFINITION.

Let 0 be as above. For c G C'°°(/, Ì ) denote by 6C the

subset of c*TM, formed by the 0C, = 0C Ð Tc,, which corresponds under r*c to

TC,0M ç è.

Define

(t) exVc:H\Qc)-*H\LM)

by (£(0) é- (expc(0T*c£(/)) and denote the image by %(c).

1.11

PROPOSITION,

(f) is bijective. Let c,d G C'°°(/, Ì ).

exp^1

ï e x p ^ e x p ^ ^ c ) Ð %(/)) - exp^(%(/) Ð %(c))

w á diffeomorphism between open sets in the Hubert Spaces

Hx(c*TM)

and

Hl(d*TM).

PROOF.

%(c) consists precisely of those e G Ç\I, M) with e(t) G expc(/)(0£(o).

This shows that (f) is a bijection.

For each t G / we form

ecrf,/ = ecr n (exP

°

T*c)"!

°

(exP

°

T*d)&d,t

and put U

0 / I

6 ^ , = 0ci/ if 0Ct#f, * 0 for all / G /. Otherwise put Öc

d

= 0 .

0Cd is an open subset of 0C, and

H\^d) = exp-'i^ic) Ð %(/)).

The map

fdc: (exp ï

ô**/)"1

ï (exp ï

T

*

c

) : 0cd - /*ÃÌ

is a fibre map,

exp^1

© expc = /£c. Hence, 1.9 applies. D

1.12

THEOREM.

The set H\I, M) of the

Hx-mapping

c: I - Ì is á Hubert

manifold; its differentiable structure is given by the natural atlas

{exp"1,

%(c); c G

C'°°(/, M)}.

PROOF.

The Charts are modelled on a proper separable Hubert Space with

typical representative H\c*TM) ss Ç\l, R").