6
THE REPRESENTATION PROBLEM
[ I
ping , of a c l o s e d i n t e r v a l ; see 4 . 1 ) whil e 3 i s t h e c l a s s of mappings
f:I Y, where X £ and Y i s m e t r i c .
The fundamental c o n c e p t i s a r e s t a t e m e n t of t h e c o n d i t i o n t h a t
th e Freche t d i s t a n c e between two mappings i s z e r o . A mapping fx'X Y i s
sai d t o be Frechet equivalent to a mapping fs:^a ~* ^ ( n o t a t i o n : ~ / ^ )
i f and only i f fo r every e 0 t h e r e i s a homeomorphism he:X1 £ XQ such
t h a t :
p { / i , fshe} e.
I n c o n s i d e r i n g t h i s concept , l e t i t be understoo d t h a t th e l e t t e r
h (with o r withou t s u b s c r i p t ) i n d i c a t e s a homeomorphism. Then fi^fa i f
and only i f for every e 0 th e following s i t u a t i o n e x i s t s :
Y
I t i s t o be note d t h a t ft (X) and fs(X) must be i n th e same m e t r i c space
before th e mappings have any chance of being Freche t e q u i v a l e n t . The diagram
shows t h a t beginnin g with any x £ X, one can go i n t o Y by means of fx,
or by followed by fQ, and th e images of x i n Y ar e w i t h i n e of
each o t h e r . Under t h e s e c i r c u m s t a n c e s , i t w i l l be s t a t e d t h a t t h e r e i s an.
approximate matching between t h e mapping s fx and / ^ . I n t h e e v e n t
p{fx, f^ey = ° fo r some he, the n t h e diagram i s s a i d to be commutative
(fx = f2h€) and the matching i s said to be exact.
I t i s eas y t o se e t h a t an e q u i v a l e n t d e f i n i t i o n i s t h e f o l l o w -
ing : fx ^ fa i f an( ^ only i f t h e r e i s a sequence, {hn}, of homeomorphisms,
hn:Xx £ XQ, n - 1, 2, 3, •, such t h a t f$hn z; f±.
The r e l a t i o n ~ i s r e a d i l y seen t o be an e q u i v a l e n c e r e l a t i o n
over th e c l a s s 3 and so p a r t i t i o n s i t i n t o mutuall y e x c l u s i v e equivalenc e
c l a s s e s , [ / ] . Each equivalency c l a s s , [ / ] , i s known as a Frechet variety,
V* Any r e p r e s e n t a t i v e of th e e q u i v a l e n c e c l a s s t h a t i s , any mapping i n
[/] i s s a i d t o be a representation of t h e F r e c h e t v a r i e t y , IT-
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