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On ∧-Mackey convergence in locally convex spaces

Published online by Cambridge University Press:  24 October 2008

Jan H. Fourie
Jan H. Fourie
Affiliation:
Department of Mathematics, Potchefstroom University, Potchefstroom 2520, South Africa
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Extract

In this note we introduce the concepts of Λ-Mackey sequence, Λ-Mackey convergence property, Λ-Schwartz family and associated Λ-Schwartz family and consider some applications of these to locally convex spaces. Hereby Λ denotes a Banach sequence space with the AK-property — the results of this paper generalize those in [4] where the case Λ = I1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the Λ-Mackey convergence property and characterize the dual Λ-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra-bornological is proved.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 495 - 500
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Fourie, J. H. and Swart, J.. Operators factoring compactly through a normal BK-space with AK. Tamkang J. Math. 13 (1982), 231–242.Google Scholar
[2]Fourie, J. H.. Quasi and Pseudo Λ-compact operators and locally convex spaces. Tamkang J. Math. 14 (1983), 171–181.,.Google Scholar
[3]Hogbe-Nlend, H.. Bornologies and Functional Analysis (North Holland, 1977).Google Scholar
[4]Jabchow, H. and Swabt, J.. On Mackey convergence in locally convex spaces. Israel J. Math. 16 (1973), 150–158.Google Scholar
[5]Jarchow, H.. Locally Convex Spaces (B. G. Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[6]Kamthan, P. K. and Gupta, M.. Sequence Spaces and Series. Lecture Notes in Pure and Applied Math., vol. 65 (Marcel Dekker, 1981).Google Scholar
[7]Lang,, K.Über spezielle Klassen von Schwartz Raümen. Arch. Math. (Basel) 28 (1977), 287–296.CrossRefGoogle Scholar
[8]Pietsch, A.. Nuclear Locally Convex Spaces (Springer-Verlag, 1972).Google Scholar
[9]Ruckle, W. H.. Sequence Spaces. Research Notes in Math., vol. 49 (Pitman Advanced Publ. Program, 1981).Google Scholar
[10]Swart, J.. Applications of Schwartz Bornologies. Math. Colloq. Univ. Cape Town 12 (1978), 33–42.Google Scholar
[11]Terzioglu, T.. Remarks on compact and infinite nuclear mappings. Math. Balkanica 2, (1972), 251–255.Google Scholar