1.3] THE PROBLEM

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I t i s now p o s s i b l e t o s t a t e t h e representation problem for

Frechet varieties:

Given one representation of a Frechet variety, find all of its

representations.

1.3 Of course, an obvious solution i s t h i s : If / i s a representa -

tion of V, then the t o t a l i t y of representations of V i s the t o t a l i t y of

mappings, g, such that / ~ g. On the other hand, the definition of Frechet

equivalence i s somewhat descriptive and often difficul t to use, so that what

one reall y desires i s an equivalent definition which i s more constructive in

character . The principa l reason for the difficult y encountered in the use

of the notion of Frechet equivalence, by the way, i s that the definition i s

in terms of an approximate rathe r than an exact matching (see 1.2). On the

othe r hand, the analyti c theory of surfaces makes t h i s d e f i n i t i o n of an

equivalence practically mandatory (see Youngs [ 14 ] ) .

The importance of the problem i s due principally to the fact that

a Frechet variety , known in terms of a particula r representation may have,

for analytic or other reasons, more favorable representations . Or as Rado

[8, p. 420] has put i t , given a p a r t i c u l a r mapping, f, there may be more

favorable mappings, g, in the c o l l e c t i o n of solution s of the r e l a t i o n

f ~ g, rphe representatio n problem, therefore, asks for suitabl e c r i t e r i a

with which to tes t the validit y of the statement / ~ g; any such criterio n

will be called an F-criterion, (see Rado" [8, p.420]).

1.4 A f i r s t simple attack on the problem might attempt to capitaliz e

directl y on the fact that i f two mappings are Frechet equivalent, then i t i s

possible to match them approximately, the degree of approximation being en-

t i r e l y within one's control . I t i s not unnatural to feel that , since the

accuracy of the approximation can be controlled , an exact matching should

be possible , thus arriving at a simple F-criterion . That i s , i f f± ^ f%

i t should be possible , one might feel , to find a homeomorphism h:X± % 1^

such that f± = fgh. This i s not the case, as the following example shows.

Suppose that I'± = I2 - I i s the set of point s 0 6: x - 1 on

the rea l l i n e . The mappings fx and fQ will be defined so as to carry

I onto i t s e l f in the following manner: