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.
@article{urn:eudml:doc:41970,
title = {On the three-space problem and the lifting of bounded sets.},
journal = {Collectanea Mathematica},
volume = {44},
year = {1993},
pages = {81-89},
zbl = {0803.46001},
mrnumber = {MR1280727},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41970}
}
Dierolf, Susanne. On the three-space problem and the lifting of bounded sets.. Collectanea Mathematica, Tome 44 (1993) pp. 81-89. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41970/