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Convergence tensor products and a strict topology

Published online by Cambridge University Press:  17 April 2009

Bernd Müller
Bernd Müller
Affiliation:
Universität Mannheim, Fakultät für Mathematik und Informatik, D-68 Mannheim, Germany.
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Abstract

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We are interested in the strict topology τ on , the set L(E, F) of all continuous linear mappings from E into a Banach space F endowed with the topology of pointwise convergence. The T3-completion of the convergence tensor product E ⊗cLc F is the set of all τ-continuous linear functionals on L(E, F) and τ is the topology of uniform convergence on the compact subsets of . In the case that E is a nuclear Fréchet space, a nuclear (DF)-space or a Banach space with the bounded approximation property the topology τ agrees with the topology of Lco (E, F).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 2 , May 1980 , pp. 281 - 301
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Antoine, Philippe, “Étude élémentaire des catégories d'ensembles structurés”, Bull. Soc. Math. Belg. 18 (1966), 387–413.Google Scholar
[2]Beattie, Ronald, “On the convergence vector space Lc(E, F) and its dual space”, Canad. Math. Bull. 21 (1978), 279–284.CrossRefGoogle Scholar
[3]Binz, E. und Keller, H.H., “Funktionenräume in der Kategorie der Limesräume”, Ann. Acad. Sci. Fenn. Ser. A I Math. 383 (1966), 1–21.CrossRefGoogle Scholar
[4]Brauner, Kalman, “Duals of Frechet spaces and a generalization of the Banach-Dieudonné theorem”, Duke Math. J. 40 (1973), 845–855.CrossRefGoogle Scholar
[5]Butzmann, H.-P., “Über die c-Reflexivität von Cc(X)”, Comment. Math. Helv. 47 (1972), 92–101.CrossRefGoogle Scholar
[6]Cook, C.H. and Fischer, H.R., “On equicontinuity and continuous convergence”, Math. Ann. 159 (1965), 94–104.CrossRefGoogle Scholar
[7]Fischer, H.R., “Limesräume”, Math. Ann. 137 (1959), 269–303.CrossRefGoogle Scholar
[8]Garling, D.J.H., “A generalized form of inductive-limit topology for vector spaces”, Proc. London Math. Soc. (3) 14 (1964), 1–28.CrossRefGoogle Scholar
[9]Grothendieck, Alexandre, Produits tensoriels topologiques et espaces nucléaires (Memoirs of the American Mathematical Society, 16. American Mathematical Society, Providence, Rhode Island, 1955).Google Scholar
[10]Jarchow, Hans, “Marinescu-Räume”, Comment. Math. Helv. 44 (1969), 138–163.CrossRefGoogle Scholar
[11]Johnson, W.B., Rosenthal, H.P. and Zippin, M., “On bases, finite dimensional decompositions and weaker structures in Banach spaces”, Israel J. Math. 9 (1971), 488–506.CrossRefGoogle Scholar
[12]Müller, Bernd, “Vervollständigungen von Limesvektorräumen” (Manuskripte, Fakultät für Mathematik und Informatik, Universität Mannheim, Mannheim, 1975).Google Scholar
[13]Müller, Bernd, “T 3-completions of convergence vector spaces”, General topology and its relations to modern analysis and algebra IV, Part B, 298–307 (Proceedings of the Fourth Prague Topology Symposium, 1976. Society of Czechoslovak Mathematicians and Physicists, Prague, 1977).Google Scholar
[14]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
[15]Schaefer, Helmut H., Topological vector spaces, Third printing corrected (Graduate Texts in Mathematics, 3. Springer-Verlag, New York, Heidelberg, Berlin, 1971).CrossRefGoogle Scholar
[16]Wiweger, A., “Linear spaces with mixed topology”, Studia Math. 20 (1961), 47–68.CrossRefGoogle Scholar