Strong duals of projective limits of (LB)-spaces

J. Bonet; Susanne Dierolf; J. Wengenroth

Czechoslovak Mathematical Journal (2002)

  • Volume: 52, Issue: 2, page 295-307
  • ISSN: 0011-4642

Abstract

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We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called “strong P-type”. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.

How to cite

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Bonet, J., Dierolf, Susanne, and Wengenroth, J.. "Strong duals of projective limits of (LB)-spaces." Czechoslovak Mathematical Journal 52.2 (2002): 295-307. <http://eudml.org/doc/30699>.

@article{Bonet2002,
abstract = {We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called “strong P-type”. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.},
author = {Bonet, J., Dierolf, Susanne, Wengenroth, J.},
journal = {Czechoslovak Mathematical Journal},
keywords = {derived projective limit functor; Retakh’s condition; weakly acyclic (LF)-spaces; derived projective limit functor; Retakh's condition; weakly acyclic (LF)-spaces},
language = {eng},
number = {2},
pages = {295-307},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strong duals of projective limits of (LB)-spaces},
url = {http://eudml.org/doc/30699},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Bonet, J.
AU - Dierolf, Susanne
AU - Wengenroth, J.
TI - Strong duals of projective limits of (LB)-spaces
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 2
SP - 295
EP - 307
AB - We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called “strong P-type”. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.
LA - eng
KW - derived projective limit functor; Retakh’s condition; weakly acyclic (LF)-spaces; derived projective limit functor; Retakh's condition; weakly acyclic (LF)-spaces
UR - http://eudml.org/doc/30699
ER -

References

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