12
THE REPRESENTATION PROBLEM
[ I

h
J
T
then one has the following commutative diagram:
On the other hand, t h i s i s not a sufficient condition, as was
o r i g i n a l l y supposed (for a h i s t o r y of t h i s erro r the reader may consul t
Youngs [ 17 ] ) , but a l l the known sufficien t condition s are arrive d at
by suitable refinements of thi s necessary condition of Kerekj arto'. Indeed,
t h i s necessary condition may be said to exhibi t a standard for desirable
F-criteria. F i r s t , an exact matching at the leve l of the range spaces i s
usuall y impossible, but the necessary condition of Kerekj arto' shows tha t
exact matching at the level of a middle space i s possible after the mappings
have been suitably factored. Second, from the level of the middle spaces on
i t i s highly important, for analytic reasons, to have the matched mappings
light,
1.8 To retur n to the necessary condition of Kerekj arto', i t i s now
easy to see how i t can be modified to obtain a certai n type of sufficien t
condition. The salient remark i s this : fx ~ f^ i f there are monotone—light
factorizations , lm± for and £m for / ^ such that ~ m^. (See
Youngs [ 11 ].)
Of course, thi s sufficient condition merely pushes back the prob-
lem of approximately matching the mappings themselves to the problem of
approximately matching monotone factors of the mappings, and so i s subject to
some, but by no means all , of the same difficultie s as the original definition
of Frechet equivalence. On the other hand, i t i s important to remark that
thi s sufficient condition i s also necessary (see Youngs [ 11 ] ) , and so r e -
duces the problem to finding F-criteri a for monotone mappings rather than gen-
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