Locally Convex Spaces and Linear Partial Differential Equations

ISBN-13: 9783642873737 / Angielski / Miękka / 2012 / 123 str.

François Treves

Opis

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."

Spis treści

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 Fréchet 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 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.