LECTURES ON HILBER T CUB E MANIFOLD S

5

5.1 . Theorem . Let X be locally compact and for each integer n 1 let Hn be a family of

homeomorphisms of X onto itself such that for each open cover U of X there exists an f G ff

n

which is U-close to id . Then we can select h

n

£ -H

n

sue/ ? fna f fne /ef t composition

h = n ^ o h n h n - r * ' h 1

defines a homeomorphism of X o/?f o /fse/f . Moreover h ca n 6e chosen as close to id as we please.

Proof . W e firs t assum e X to be compact . Fo r an y tw o homeomorphism s f,g : X -* X defin e

p(f,g ) = d f , g ) + d ( r V

1

) .

If #(X ) is th e spac e o f al l homeomorphism s o f X ont o itself , endowe d wit h th e usua l sup-nor m

metric , the n th e reade r ca n chec k tha t p define s a n equivalen t metri c fo r HiX) whic h is

oo

complete . Usin g thi s complet e metri c we ca n selec t h

n

G tfn so tha t th e sequenc e { h

n

• • • h-j } -

is Cauchy , an d therefor e

h = lim h

n

• * • h i

is a homeomorphism . Certainl y h ca n be chose n as clos e to id as we please . Thi s take s car e o f

the compac t case .

We ca n reduc e th e non-compac t cas e to th e compac t cas e as follows . Fo r X non-compac t le t

X = X U {«} , th e one-poin t compactificatio n o f X , an d defin e

p:U(X)-+H(X)

by lettin g ^?(h) = h , o n X , an d ^lh)(o©) = oo. if # ( x ) is give n th e compact-ope n topolog y an d

fW(X ) is as above , the n th e reade r ca n chec k tha t p is a n embedding . Usin g th e compac t cas e we

ca n selec t h

n

€ Hn so tha t

h = lim ^(h n») • • -sp(hi )

n-*oo i

define s a n elemen t o f #(X) . The n h = ^ ~ ' ( h ) fulfill s ou r requirements . If th e h

n

ar e selecte d

sufficientl y clos e to id , the n h ca n be mad e as clos e to id as we please . •