Glasnik Matematicki, Vol. 43, No.2 (2008), 397-422.

ON HEREDITARY REFLEXIVITY OF TOPOLOGICAL VECTOR SPACES. THE DE RHAM COCHAIN AND CHAIN SPACES

Ju. T. Lisica

Abstract.   For proving reflexivity of the spaces of de Rham cohomology and homology of C^∞-manifolds the author considers the notion of hereditary reflexivity as well as the notion of dual hereditary reflexivity of locally convex topological vector spaces which is interesting in itself. Complete barrelled nuclear spaces with complete nuclear duals turn out to be hereditarily reflexive. The Pontryagin duality in locally convex topological spaces is also considered.

2000 Mathematics Subject Classification.   46A03, 46A04, 55N07, 55P55.

Key words and phrases.   De Rham cohomology, currents, nuclear spaces, hereditary reflexivity, dual hereditary reflexivity, Pontryagin duality.

DOI: 10.3336/gm.43.2.12

References:

1. N. Bourbaki, Espaces Vectoriels Topologiques, Hermann & Cie, Éditeur, Paris, 1953, 1955.
   MathSciNet

2. J. Dieudonné, La dualité dans les espaces vectoriels topologiques, Ann., Sci. École Norm. Sup. (3) 59 (1942), 107-139.
   MathSciNet     Numdam

3. A. Grothendieck, Sur la complétion du dual d'un espace vectoriel localement convexe, C. R. Acad. Sci. Paris 230 (1950), 605-606.
   MathSciNet

4. A. Grothendieck, Sur une notion de produit tensoriel topologique d'espaces vectoriels topologiques, et une classe remarquable d'espaces vectoriels liée à cette notion, C. R. Acad. Sci. Paris 233 (1951), 1556-1558.
   MathSciNet

5. A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasil. Math. 3 (1954), 57-123.
   MathSciNet

6. S. Kaplan, Extensions of Pontryagin duality, I. Infinite products, Duke Math. J. 15 (1948), 649-658; II. Direct and inverse sequences, ibid. 17 (1950), 419-435.
   MathSciNet
   MathSciNet

7. Y. Komura, Some examples on linear topological spaces, Math. Ann. 153 (1964), 150-162.
   MathSciNet     CrossRef

8. Ju. T. Lisica, Main theorems of strong shape theory of compact Hausdorff spaces, Vestnik Ross. Univ. Druzhby Narodov 7 (2000), 63-93 (in Russian).

9. Ju. T. Lisica, Coherent homotopy, homology, cohomology and strong shape theory, Dissertation, Moscow, 2001 (in Russian).

10. Ju. T. Lisica, Theory of spectral sequences. II, Fundam. Prikl. Mat 11 (2005) 117-149.
    MathSciNet     CrossRef

11. V. V. Marchenko, Commutative cochains of germs of differentiable forms of compacta in Euclidean spaces, B. Sc. thesis, Ross. Univ. Druzhby Narodov, 2000 (in Russian).

12. S. Mardesic, Strong Shape and Homology, Springer-Verlag, Berlin, 2000.
    MathSciNet

13. S. A. Morris, Pontryagin duality and the structure of locally compact abelian groups, London Mathematical Society Lecture Note Series 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977.
    MathSciNet

14. A. Pietsch, Nukleare lokalkonvexe Räume, Akademie-Verlag, Berlin, 1965.
    MathSciNet

15. L. S. Pontryagin, Continuous groups, Nauka, Moscow, 1973.
    MathSciNet

16. D. A. Raykov, Some linear-topological properties of the spaces D and D' (in Russian), Appendix 2 in the book A.P. Robertson and W.R. Robertson, Topological Vector Spaces, Mir, Moscow, 1967, 238-250.

17. G. de Rham, Variétées Différentiables. Formes, courants, formes harmoniques, Hermann & Cie, Éditeur, Paris, 1955.
    MathSciNet

18. A. P. Robertson and W. R. Robertson, Topological Vector Spaces, Cambridge Tracts in Mathematics and Mathematical Physics 53}, Cambridge University Press, New York, 1964.
    MathSciNet

19. H. H. Schaefer, Topological Vector Spaces, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966.
    MathSciNet

20. W. Slowikowski, Fonctionelles linéaires dans des réunions dénombrables d'espaces de Banach réflexifs, C. R. Acad. Sc. Paris Sér A-B \textbf{262} (1966), A870-A872.
    MathSciNet

21. M. F. Smith, The Pontrjagin duality theorem in linear spaces, Ann. of Math. (2) 56 (1952), 248-253.
    MathSciNet     CrossRef