Locally Convex Spaces and Linear Partial Differential Equations (Grundlehren der mathematischen Wissenschaften (146))

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. Front Matter....Pages I-XII Front Matter....Pages 1-1 The Spectrum of a Locally Convex Space....Pages 3-14 The Natural Fibration over the Spectrum....Pages 15-24 Epimorphisms of Fréchet Spaces....Pages 25-36 Existence and Approximation of Solutions to a Functional Equation....Pages 37-42 Translation into Duality....Pages 43-54 Front Matter....Pages 55-55 Applications of the Epimorphism Theorem....Pages 57-71 Applications of the Epimorphism Theorem to Partial Differential Equations with Constant Coefficients....Pages 72-82 Existence and Approximation of Solutions to a Linear Partial Differential Equation....Pages 83-91 Existence and Approximation of Solutions to a Linear Partial Differential Equation....Pages 92-104 Back Matter....Pages 105-123