For the past two decades it has been known that many of the
function spaces which arise naturally in the investigation of problems
involving non-linear partial differential operators possess natural
Banach manifold differentiable structures, and that C non-linear
differential operators often induce differentiable maps between
appropriate function space manifolds. In this way, the tools of
differential calculus and infinite-dimensional differential topology
can be brought to bear on such problems. However, such tools by
themselves are rarely adequate, and in situations where infinite-dimen-
sional differential topology can be successfully applied it is usually
necessary to use additional properties which function spaces and
operator-induced morphisms possess, but which have no abstract equiva-
lents in Banach manifold differential topology. indeed, some global
analysts insist that it is useless to consider an abstract model at
all, that global analysts should forget about abstractions and work
exclusively with function spaces and differential operators.
However, the purpose of this monograph is to take a third ap-
proach: while conceding that Banach manifold differential topology is
too weak an abstract setting for global analysis* I present herein a
more highly structured category whose objects seem to possess many of
the properties relevant to concrete applications. Many function spaces
will be seen to be examples of these new abstract objects; and as an
illustration of the suitability of this model, several abstract
theorems will be presented which imply the basic results of Palais and
Smale in non-linear variational calculus.
The key to this theory is the bw* space. The simplest (though
not most elegant) way to describe a bw* space is as the dual space
VI
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