Locally Convex Spaces and Linear Partial Differential Equations I. The Spectrum of a Locally Convex Space I. The Spectrum of a Locally Convex Space II. The Natural Fibration over the Spectrum III. Epimorphisms of Fréchet Spaces IV. Existence and Approximation of Solutions to a Functional Equation V. Translation into Duality II. Applications to Linear Partial Differential Equations VI. Applications of the Epimorphism Theorem VII. Applications of the Epimorphism Theorem to Partial Differential Equations with Constant Coefficients VIII. Existence and Approximation of Solutions to a Linear Partial Differential Equation IX. Existence and Approximation of Solutions to a Linear Partial Differential Equation Appendix A: Two Lemmas about Fréchet Spaces Appendix B: Normal Hilbert Spaces of Distributions Appendix C: On the Nonexistence of Continuous Right Inverses Main Definitions and Results Concerning the Spectrum of a Locally Convex Space Some Definitions in PDE Theory Bibliographical References. 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 vs 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.