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A note on Taskinen's counterexamples on the problem of topologies of Grothendieck

Published online by Cambridge University Press:  20 January 2009

Jose Bonet  and
Antonio Galbis
Jose Bonet
Affiliation:
Departamento de Matemáticas, E.T.S. Arquitectura, Universidad Politécnica de Valencia, 46022 Valencia, Spain
Antonio Galbis
Affiliation:
Departamento de Análisis, Facultad de Matemáticas, Universidad de Valencia, 46071 Burjasot (Valencia), Spain
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By the work of Taskinen (see [4, 5]), we know that there is a Fréchet space E such that Lb(E, l2) is not a (DF)-space. Moreover there is a Fréchet–Montel space F such that is not (DF). In this second example, the duality theorem of Buchwalter (cf. [2, §45.3]) can be applied to obtain that and hence is a (gDF)-space (cf. [1, Ch. 12 or 3, Ch. 8]). The (gDF)-spaces were introduced by several authors to extend the (DF)-spaces of Grothendieck and to provide an adequate frame to consider strict topologies.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 2 , June 1989 , pp. 281 - 283
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

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4.Taskinen, J., Counterexamples to “Probleme des topologies” of Grothendieck, Ann. Acad. Sci. Fenn. Ser. A, I. Math. Dissertationes 63 (1986).Google Scholar
5.Taskinen, J., The projective tensor product of Fréchet-Montel spaces, Studia Math. 91 (1988), 17–30.CrossRefGoogle Scholar