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Proceedings of the American Mathematical Society

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Some problems on $ B$-completeness


Author: T. K. Mukherjee
Journal: Proc. Amer. Math. Soc. 46 (1974), 367-374
MSC: Primary 46A30
DOI: https://doi.org/10.1090/S0002-9939-1974-0367613-1
MathSciNet review: 0367613
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Abstract: In this paper, we give examples to show the following:

1. The product of two $ B$-complete spaces is not necessarily $ {B_r}$-complete.

2. The Mackey dual of a strict LF-space is not necessarily $ {B_r}$-complete.

3. A separable, reflexive, and strict LF-space is not, in general, $ {B_r}$-complete. The second point has reference to a problem of Dieudonné and Schwartz which asks essentially whether the Mackey dual of a strict LF-space is $ B$-complete and which was answered in the negative by Grothendieck.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0367613-1
Keywords: $ B$-complete, $ {B_r}$-complete, sequentially closed subspace, LF-space
Article copyright: © Copyright 1974 American Mathematical Society