STRICTLY POSITIVE FUNCTIONALS ON VECTOR LATTICES By JAN A. VAN CASTEREN [Received 12 April 1977—Revised 9 January 1978] 0. Introduction The problem we study is the following. Let E be a topological vector lattice. Does there exist a continuous linear functional <p on E such that <p( | u |) is positive for each non-zero u in E ? Several authors have worked on this problem in the case where E is the space C(X) of all continuous complex functions on a compact Hausdorfif space X. For example, see Kelley [5], where measures on Boolean algebras are considered. Gaifman [3] showed that the countable chain condition is not sufficient for the existence of a bounded strictly positive Radon measure on X. Rosenthal [9, Theorem 4.5.b] showed that C(X) carries a strictly positive functional if and only if its dual C(X)' contains a weakly compact total subset. Another criterion for the existence of strictly positive functionals can be found in a paper by Hebert and Lacey [4]. Moore [7] also gives necessary and sufficient conditions for the existence of a strictly positive functional on a normed vector lattice. All the results in C(X) have been generalized to non-compact X by the author in [14, Chapter IV]. Our aim is to develop a theory along the lines of the cited references in the case where C(X) is replaced with an arbitrary topological vector lattice; see Schaefer [10, Chapter V] for the precise definitions. 1. Notation, conventions and preliminary results We consider a partially ordered complex vector space E with positive cone C. A subset 8 of E is order convex if for each pair (u lt u 2 ) in S x S for which u x ^ u 2 the order interval [u v u 2 ], defined by [u v u 2 ] = {u E E: M X < u ^ u 2 }, is a subset of S. In particular we are interested in order-convex subcones of C. A subcone S of C is always supposed to be convex. It is order convex if and only if for each u in S the order interval [0, u] is contained in S. We assume that C is generating: Proe. London Math. Soc. (3) 39 (1979) 51-72