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 . •
