J. Aust. Math. Soc. 75 (2003), 163–180 PROJECTIVE DESCRIPTIONS OF THE (LF)-SPACES OF TYPE L B. p.A/; F/

Let 1 p < C1 or p D 0 and let A D.an/n be an increasing sequence of strictly positive weights on I. Let F be a Fréchet space. It is proved that if p.A / satisfies the density condition of Heinrich and a certain condition (Ct) holds, then the (LF)-space LBi.p.A/; F / is a topological subspace of Lb.p.A/; F/. It is also proved that these conditions are necessary provided F D q.A / or F contains a complemented copy of lq with 1 < p q < C1. 2000 Mathematics subject classification: primary 46A13, 46A32, 46A45. Keywords and phrases: spaces of linear continuous maps, echelon and co-echelon spaces of order p, projective description. The space LB.E; F / of all linear bounded maps between two locally convex spaces E and F has been considered and studied under different points of view by severa