On the three-space-problem and the lifting of bounded sets,

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 really different possibilities of defining the notion of a dual Fréchet space. If in a Three-spaceproblem 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.