# 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

## Access Full Article

top## Abstract

top## How to cite

topDierolf, 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 -

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.