It is hardly an exaggeration to say that, if the study of general
topolog- ical vector spaces is justified at all, it is because of the
needs of distribu- tion and Linear PDE * theories (to which one may add
the theory of convolution in spaces of hoi om orphic functions). The
theorems based on TVS ** theory are generally of the "foundation"
type: they will often be statements of equivalence between, say, the
existence - or the approx- imability -of solutions to an equation Pu =
v, and certain more "formal" properties of the differential operator P,
for example that P be elliptic or hyperboJic, together with properties
of the manifold X on which P is defined. The latter are generally
geometric or topological, e. g. that X be P-convex (Definition 20. 1).
Also, naturally, suitable conditions will have to be imposed upon the
data, the v's, and upon the stock of possible solutions u. The effect of
such theorems is to subdivide the study of an equation like Pu = v into
two quite different stages. In the first stage, we shall look for the
relevant equivalences, and if none is already available in the
literature, we shall try to establish them. The second stage will
consist of checking if the "formal" or "geometric" conditions are
satisfied.