Three-space-problem for inductively (semi)-reflexive locally convex spaces

Three-space-stability of inductively (semi)-reexive and some re- lated classes of locally convex spaces is considered. It is shown that inductively (semi)-reexive spaces behave more regularly than (semi)-reexive spaces in that sense. Let (E;t) be a Hausdor locally convex space (l.c.s.) with the topological dual E0; there exist several topologies on E0 (the weak topology (E0;E), the topology (E0;E) of uniform convergence on compact and absolutely convex sets, Mackey topology (E0;E), the strong topologyb(E0;E) and others). The so-called inductive topology TE0 on E0 was introduced in (3) and (5) as the inductive-limit topology of the Banach spaces E0V , where V runs through a zero-neighborhood basis of (E;t) formed by closed and absolutely convex sets. Here E0V = S n2NnV is equipped with the norm having V as the unit ball. The zero-neighborhood basis of TE0 is formed by all absolutely convex subsets of E0 that absorb all t-equicontinuous subsets. This topology is the strongest locally convex topology on E0 for which all t-equicontinuous subsets are bounded. Particularly, it is