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