Locally Convex Spaces and Linear Partial Differential Equations

I. The Spectrum of a Locally Convex Space.- I. The Spectrum of a Locally Convex Space.- 1. Seminorms on a vector space.- 2. The spectrum of a locally convex space.- 3. Polar. Bipolar.- 4. Continuous linear mappings.- II. The Natural Fibration over the Spectrum.- 5. The natural fibration over the spectrum.- 6. The irreducible subsets of the spectrum interpreted as the equivalence classes of continuous linear mappings with dense image.- 7. Spectra of inductive and projective limits of locally convex spaces.- III. Epimorphisms of Frechet Spaces.- 8. Equicontinuous subsets of the spectrum.- 9. Barrels. Barrelled spaces.- 10. The epimorphism theorem.- 11. Criteria of presurjectivity.- 12. The closed graph theorem.- IV. Existence and Approximation of Solutions to a Functional Equation.- 13. The canonical extension of an essentially univalent linear mapping.- 14. Existence and approximation of solutions to a functional equation.- V. Translation into Duality.- 15. The seminorms "absolute value of a linear functional".- 16. Lower star and transpose.- 17. Duality in relation with existence and approximation of solutions to a functional equation.- II. Applications to Linear Partial Differential Equations.- VI. Applications of the Epimorphism Theorem.- 18. A classical theorem of E. Borel.- 19. Estimates in Sobolev spaces leading to the existence of solutions to a linear PDE.- 20. P-convexity and semiglobal solvability.- 21. Remarks on P-convexity and semiglobal solvability.- VII. Applications of the Epimorphism Theorem to Partial Differential Equations with Constant Coefficients.- 22. On certain Frechet spaces of distributions.- 23. Existence of solutions to a linear PDE with constant coefficients.- VIII. Existence and Approximation of Solutions to a Linear Partial Differential Equation.- I. General differential operators.- 24. Approximation of solutions to the homogeneous equation by C? solutions.- 25. Existence and approximations of solutions to the inhomogeneous equation.- 26. P-Runge domains and relative P-convexity.- IX. Existence and Approximation of Solutions to a Linear Partial Differential Equation.- II. Differential operators with constant coefficients.- 27. Spaces of polynomials, of formal power series, of exponential-polynomials, of entire functions of exponential type...- 28. Existence of solutions in the spaces of polynomials, of formal power series, of exponential-polynomials, of entire functions of exponential type.- 29. Existence and approximation of solutions in the space of entire functions.- 30. Further theorems of existence and approximation of solutions.- Appendix A: Two Lemmas about Frechet 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.