Math. Proc. Camb. Phil. Soc. (1983), 93, 307-314 307 Printed in Great Britain Topological decompositions of the duals of locally convex operator spaces BY D. J. FLEMING AND D. M. GIARRUSSO St Lawrence University, Canton, New York 13617 (Received 13 July 1982) 1. Introduction If Z and E are Hausdorff locally convex spaces (LCS) then by L b (Z, E) we mean the space of continuous linear maps from ZtoE endowed with the topology of uniform con- vergence on the bounded subsets of Z. The dual L b (Z, E)' will always carry the topology of uniform convergence on the bounded subsets of L b (Z,E). If K(Z,E) is a linear subspace of L(Z, E) then K b (Z, E) will be used to denote K(Z, E) with the relative topology and K b (Z, E)" will mean the dual of K b (Z, E)' with the natural topology of uniform convergence on the equicontinuous subsets of K b (Z, E)'. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies. We shall refer to a continuous linear map from Z to E as quasi-compact if it takes bounded subsets of Z into relatively compact subsets of E. The space of all quasi- compact (respectively, compact) maps from Z to E is denoted by K(Z, E) (respectively, C(Z, E)). HE is quasi-complete it is routine to verify that K(Z, E) is closed in L b (Z, E). In general C(Z, E) is a proper subset o£K(Z, E); however, from Grothendieck(5) these spaces coincide whenever Z is quasi-normable and E is Fr6chet. (See also (9).) The purpose of this paper is to investigate the structure oiL b (Z, E)' and K b (Z, E)" for Z and E Hausdorff LCS. In Section 3 we introduce the notion of a fully equicon- tinuous quasi-compact expansion of the identity map on E. Given such an expansion of id E we demonstrate the existence of a continuous projection from L b (Z, E)' onto the closed subspace K b (Z, E) x whenever E is quasi-complete. (For Z, E, Banach spaces this provides a generalization of theorems of Hennefeld ((7); 1-2, 1-3)). If in addition Z is quasi-barrelled and E semi-reflexive we obtain a topological decomposition of K b (Z,E)" where one of the factors is isomorphic to L b (Z,E). We show directly that if E is quasi-complete and K b (Z, E) is semi-reflexive then L(Z, E) = K(Z, E). For Z quasi-barrelled we obtain necessary and sufficient conditions for L b (Z, E) to be semi- reflexive. 2. Notation and terminology For the most part we follow the conventions and notation of Bourbaki(i). In particular if Z and E are Hausdorff LCS then a base for the neighbourhoods of zero for the topology on L b (Z, E) is given by all sets of the form T(A, U) = {ueL(Z,E)\u(A) s U} where A is bounded in Z and U is a neighbourhood of 0 in E. If q is a continuous seminorm on E and A is bounded in Z then by q A we mean the seminorm on L(Z, E) given by q A (u) = sup<7(w(z)). The collection of all such q A is a generating family of seminorms for the topology of L b (Z, E). For M an equicontinuous subset of L(Z, E)