# On the three-space problem and the lifting of bounded sets.

Collectanea Mathematica (1993)

• Volume: 44, Issue: 1-2-3, page 81-89
• ISSN: 0010-0757

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## Abstract

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We exhibit a general method to show that for several classes of Fréchet spaces the Three-space-problem fails. This method works for instance for the class of distinguished Fréchet spaces, for Fréchet spaces with the density condition and also for dual Fréchet spaces (which gives a negative answer to a question of D. Vogt). An example of a Banach space, which is not a dual Banach space but the strong dual of a DF-space, shows that there are two real different possibilities of defining the notion of a dual Fréchet space. If in a Three-space-problem the corresponding quotient map is assumed to lift bounded sets, we obtain partial positive answers. Finally, we give this property of lifting bounded sets a special treatment.

## How to cite

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Dierolf, Susanne. "On the three-space problem and the lifting of bounded sets.." Collectanea Mathematica 44.1-2-3 (1993): 81-89. <http://eudml.org/doc/41970>.

@article{Dierolf1993,
abstract = {We exhibit a general method to show that for several classes of Fréchet spaces the Three-space-problem fails. This method works for instance for the class of distinguished Fréchet spaces, for Fréchet spaces with the density condition and also for dual Fréchet spaces (which gives a negative answer to a question of D. Vogt). An example of a Banach space, which is not a dual Banach space but the strong dual of a DF-space, shows that there are two real different possibilities of defining the notion of a dual Fréchet space. If in a Three-space-problem the corresponding quotient map is assumed to lift bounded sets, we obtain partial positive answers. Finally, we give this property of lifting bounded sets a special treatment.},
author = {Dierolf, Susanne},
journal = {Collectanea Mathematica},
keywords = {Problema de tres espacios; Espacios vectoriales topológicos; Espacios de Frechet; dual Banach-spaces; three-space-problem; distinguished Fréchet spaces; Fréchet spaces with the density condition; dual Fréchet spaces; strong dual of a DF-space; quotient map; lifting bounded sets},
language = {eng},
number = {1-2-3},
pages = {81-89},
title = {On the three-space problem and the lifting of bounded sets.},
url = {http://eudml.org/doc/41970},
volume = {44},
year = {1993},
}

TY - JOUR
AU - Dierolf, Susanne
TI - On the three-space problem and the lifting of bounded sets.
JO - Collectanea Mathematica
PY - 1993
VL - 44
IS - 1-2-3
SP - 81
EP - 89
AB - We exhibit a general method to show that for several classes of Fréchet spaces the Three-space-problem fails. This method works for instance for the class of distinguished Fréchet spaces, for Fréchet spaces with the density condition and also for dual Fréchet spaces (which gives a negative answer to a question of D. Vogt). An example of a Banach space, which is not a dual Banach space but the strong dual of a DF-space, shows that there are two real different possibilities of defining the notion of a dual Fréchet space. If in a Three-space-problem the corresponding quotient map is assumed to lift bounded sets, we obtain partial positive answers. Finally, we give this property of lifting bounded sets a special treatment.
LA - eng
KW - Problema de tres espacios; Espacios vectoriales topológicos; Espacios de Frechet; dual Banach-spaces; three-space-problem; distinguished Fréchet spaces; Fréchet spaces with the density condition; dual Fréchet spaces; strong dual of a DF-space; quotient map; lifting bounded sets
UR - http://eudml.org/doc/41970
ER -

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