Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-13T21:52:15.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Closed graph theorems for generalized inductive limit topologies

Published online by Cambridge University Press:  24 October 2008

W. Ruess
W. Ruess
Affiliation:
Universität Bonn
Get access Rights & Permissions [Opens in a new window]

Summary

The object of this and a subsequent paper is to investigate the locally convex structure of several strict topologies that are generalizations of R. C. Buck's strict topology β on C(S), S locally compact Hausdorff. If the topology τ of a locally convex space (lcs) (X, τ) is any of these strict topologies, then it is localizable on every absorbing disc T in X, i.e. it is the finest locally convex topology on X agreeing with τ on T. Topologies of this kind are said to be (L)-topologies. As our main tools for the analysis of the structure of strict topologies, we deduce in this paper several closed graph theorems for spaces of type (L). In particular, it is shown that every semi-Montel lcs with a fundamental sequence of bounded sets and every Bτ-complete Schwartz space belongs to the class Bτ(L) of all lcs Y with the property that every closed linear map from any (L)-space X into Y is continuous. Further closed graph theorems are established and many of the known closed graph theorems are deduced as special cases of our results. Moreover, the problem of Bτ-completeness of locally convex spaces belonging to Bτ(L) is considered.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 1 , July 1977 , pp. 67 - 83
Copyright
Copyright © Cambridge Philosophical Society 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Adasch, N.Tonnelierte Räume und zwei Sätze von Banach. Math. Ann. 186 (1970), 209–214.CrossRefGoogle Scholar
(2)Baernstein, A.II. Representation of holomorphic functions by boundary integrals. Trans. Amer. Math. Soc. 160 (1971), 27–37.CrossRefGoogle Scholar
(3)Buck, R. C.Operator algebras and dual spaces. Proc. Amer. Math. Soc. 3 (1952), 681–687.CrossRefGoogle Scholar
(4)Buck, R. C.Bounded continuous functions on a locally compact space. Michigan Math. J. 5 (1958), 95–104.CrossRefGoogle Scholar
(5)Busby, R. C.Double centralizers and extensions of C*-algebras. Trans. Amer. Math. Soc. 132 (1968), 79–99.Google Scholar
(6)Collins, H. S.Strict, weighted, and mixed topologies and applications. Advances in Math. 19 (1976), 207–237.CrossRefGoogle Scholar
(7)De Wilde, M.Réseaux dans les espaces linéaires à semi-normes. Mém. Soc. Royale Sci. Liège 18 (2) (1969).Google Scholar
(8)Eberhardt, V. Durch Graphensätze definierte lokalkonvexe Räume, Dissertation, München 1972.Google Scholar
(9)Eberhardt, V.Der Graphensatz von A. P. und W. Robertson für s-Räume. Manuscripts math. 4 (1971), 255–262.CrossRefGoogle Scholar
(10)Fremlin, D. H., Garling, D. J. H. and Haydon, R. G.Bounded measures on topological spaces. Proc. London Math. Soc. (3) 25 (1972), 115–136.CrossRefGoogle Scholar
(11)Garling, D. J. H.A generalized form of inductive-limit topology for vector spaces. Proc. London Math. Soc. 14 (3) (1964), 1–28.CrossRefGoogle Scholar
(12)Grothendieck, A.Sur les espaces (F)-et (DF). Summa Brasil Math. 3 (1954), 57–122.Google Scholar
(13)Horvath, J.Topological vector spaces and distributions, I. Addison-Wesley Publ. Comp. (1969).Google Scholar
(14)Iyahen, S. O.On the closed graph theorem. Israel J. Math. 10 (1971), 96–105.CrossRefGoogle Scholar
(15)Iyahen, S. O.A closed graph theorem. Proc. Amer. Math. Soc. 53 (1975), 165–166.CrossRefGoogle Scholar
(16)Köthe, G.Homomorphismen von (F)-Räumen. Math. Z. 84 (1964), 219–221.CrossRefGoogle Scholar
(17)Köthe, G.Topologische lineare Räume I. (Springer, Berlin, Heidelberg, N.Y. 1966).CrossRefGoogle Scholar
(18)Komura, Y.On linear topological spaces. Kumamoto J. Sci. Ser. A 5 (1960–1962), 148–157.Google Scholar
(19)McIntosh, A.On the closed graph theorem. Proc. Amer. Math. Soc. 20 (1969), 397–404.CrossRefGoogle Scholar
(20)Persson, A.A remark on the closed graph theorem in locally convex vector spaces. Math. Scand. 19 (1966), 54–58.CrossRefGoogle Scholar
(21)Powell, M. H.On Komura's closed graph theorem. Trans. Amer. Math. Soc. 211 (1975), 391–426.Google Scholar
(22)Ptak, V.Completeness and the open mapping theorem. Bull. Soc. Math. France 86 (1958), 41–74.Google Scholar
(23)Robertson, A. P. and Robertson, W.On the closed graph theorem. Proc. Glasgow Math. Assoc. 3 (1956), 9–12.CrossRefGoogle Scholar
(24)Robertson, W.On the closed graph theorem and spaces with webs. Proc. London Math. Soc. (3) 24 (1972), 692–738.CrossRefGoogle Scholar
(25)Roelcke, W.On the finest locally convex topology agreeing with a given topology on a sequence of absolutely convex sets. Math. Ann. 198 (1972), 57–80.CrossRefGoogle Scholar
(26)Ruess, W.A Grothendieck representation for the completion of cones of continuous semi-norms. Math. Ann. 208 (1974), 71–90.CrossRefGoogle Scholar
(27)Ruess, W.Generalized inductive limit topologies and barrelledness properties. Pacific J. Math. 63 (1976), 499–516.CrossRefGoogle Scholar
(28)Ruess, W.On the locally convex structure of strict topologies (to appear in Math. Z.).Google Scholar
(29)Ruess, W.Sur quelques problèmes de topologies strictes. C.R. Acad. Sci. Paris 283 (1976), A603–A606.Google Scholar
(30)Sentilles, F. D.Bounded continuous functions on a completely regular space. Trans. Amer. Math. Soc. 168 (1972), 311–336.CrossRefGoogle Scholar
(31)Sentilles, F. D. and Taylor, D. C.Factorization in Banach algebras and the general strict topology. Trans. Amer. Math. Soc. 142 (1969), 141–152.CrossRefGoogle Scholar
(32)Taylor, D. C.The strict topology for double centralizer algebras. Trans. Amer. Math. Soc. 150 (1970), 633–643.CrossRefGoogle Scholar
(33)Treves, F.Locally convex spaces and linear partial differential equations (Springer, Berlin, Heidelberg N.Y. 1967).CrossRefGoogle Scholar
(34)Valdivia, M.Sobre el teorema de la gráfica cerrada. Collect. Math. 22 (1971), 3–24.Google Scholar
(35)Valdivia, M.Mackey convergence and the closed graph theorem. Arch. Math. 25 (1974), 649–656.CrossRefGoogle Scholar
(36)Valdivia, M.The space of distributions ′(Ω) is not Br-complete. Math. Ann. 211 (1974), 145–149.CrossRefGoogle Scholar
(37)Valdivia, M. Some new results on the closed graph theorem, preprint 1975.Google Scholar
(38)Webb, J. H.Sequential convergence in locally convex spaces. Proc. Cambridge Philos. Soc. 64 (1968), 341–364.CrossRefGoogle Scholar