On the three space problems for countable barrelendness

Abstract

Let F be a closed subspace of a Hausdorff locally convex space E such that F and the Hausdorff quotient E/F enjoy a property (M). Does the whole space E enjoy (M)?.

This problem is called "the three-space-problem" and it is a common problem in the framework of twisted exact sequences {see I3I). It is been already considered by several authors, e.g. IPBI, I4I.

ln the present paper we shall provide a negative answer to the problem for IP -quasi-barrelledness, lP Є {Xo, l∞, c, co} and for df spaces. We shall also supply a thoroughly positive answer for l∞-barrelledness and a partial affirmative answer for Co-barrelledness.