Math. Proc. Camb. Phil. Soc. (1984), 96, 321 321 Printed in Great Britain Projections and embeddings of locally convex operator spaces and their duals BY JAN H. FOURIE AND WILLIAM H. RUCKLE Potchefstroom University, Potchefstroom 2520, South Africa and Clemson University, Clemson, SC 29631, U.S.A. {Received 9 September 1983; revised 16 February 1984) Abstract Let E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space K b {E, F) (of quasi-compact operators) is a complemented subspace of the dual space o£L b {E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for L b (E, F) to be semi-reflexive. 1. Introduction We consider operators on Hausdorff locally convex spaces; as usual, L b (E,F) will denote the space of continuous linear operators L(E, F) which is endowed with the topology of uniform convergence on the bounded sets of E. Similarly H b (E, F) will denote the vector subspace H(E, F) oiL b (E, F) on which the induced subspace topology is denned. The dual H b (E, F)' will always carry the topology of uniform convergence on the bounded subsets of H b (E, F), whereas H b (E, F)" will always be endowed with the topology of uniform convergence on the equicontinuous subsets of H b (E,F)'. Neighbourhoods of the origin in H b (E, F) are denoted by M(A, V), where A is bounded in E and V is a neighbourhood of the origin in F. An element of L b (E, F) is called quasi-compact if it takes bounded subsets of E into precompact subsets of F. The space of all quasicompact operators from E into F is denoted by K{E, F) (or K(E) if E = F). In [2] the structures o£L b (E, F)' and K b (E, F)" are investigated. The notion of 'fully equicontinuous quasi-compact expansion of the identity' is introduced and it is demonstrated that if such an expansion for the identity on a quasi-complete locally convex space F exists, then there is a continuous projection from L b (E,F)' onto a subspace of L b {E,F)' which is isomorphic to K b (E,F)'. This provides generalizations of results in [3] and [5] where similar results for Banach spaces were proved. If in addition E is quasi-barrelled and F is semi-reflexive, the authors of [2] obtain a topological decomposition of K b (E, F)" where one of the factors is iso- morphic to L b (E,F). These results are applied to obtain necessary and sufficient conditions for L b (E, F) to be semi-reflexive. The purpose of this paper is to establish significant generalizations of the results in [2] by introducing the q.c.a.p. (or s.q.c.a.p.) condition (cf. section 2) on F (or E). Once we have introduced the projection mapping on L b (E, F)', we shall only indicate to the reader how the hypotheses of [2] can be adjusted.