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 .
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