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A Köthe space which has a continuous norm but whose bidual does not. (English) Zbl 0724.46008

S. Dierolf and V. Moscatelli gave an example of a Fréchet space with a continuous norm but whose bidual does not admit a continuous norm. The authors construct in an elementary fashion even a Köthe space with these properties.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46A04 Locally convex Fréchet spaces and (DF)-spaces
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References:

[1] K. D. Bierstedt etJ. Bonet, La condition de densité et les espaces échelonnés distingués. C. R. Acad. Sci. Paris303, 459-462 (1986).
[2] K. D. Bierstedt, R. G. Meise andW. H. Summers, Köthe sets and Köthe sequence spaces. In: Functional Analysis, Holomorphy and Approximation Theory, Math. Studies71, 27-91, Amsterdam 1982. · Zbl 0504.46007
[3] S. Dierolf andV. B. Moscatelli, A Fréchet space which has continuous norm but whose bidual does not. Math. Z.191, 17-21 (1986). · Zbl 0561.46001 · doi:10.1007/BF01163606
[4] G.Köthe, Topologische lineare Räume I. Berlin-Göttingen-Heidelberg 1960.
[5] D. Vogt, On two classes of (F)-spaces. Arch. Math.45, 255-266 (1985). · Zbl 0621.46001 · doi:10.1007/BF01275578
[6] D.Vogt, Distinguished Köthe spaces. To appear in Math. Z.
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