Linear Operators Whose Domain Is Locally Convex

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on 5 separates the points of S. If X is a Banach space and T: X -* F is a continuous linear operator, then T is quasi-convex if T(U) is quasiconvex, where U is the unit ball of X. In the case when T is compact, T(U) is quasi-convex if and only if it is affinely homeomorphic to a subset of a locally convex space. This is immediate, since the topology on T(U) can be induced by the family of affine functionals vanishing at 0. It is also equivalent to the condition that 0 has a base of convex neighbourhoods in T(U); this is proved by constructing on the linear span of T(U) the finest vector topology y agreeing with the given topology on T(U). Then y is locally convex—this follows from results in (21, p. 51). In (16) Peck and Waelbroeck ask whether every compact convex set is locally convex. An equivalent question is whether every compact operator is quasi-convex. This question has recently been resolved negatively by J. W. Roberts (unpublished). We obtain some partial results here (for results on this problem in a different direction, see (9)). If X is reflexive, every bounded operator is quasi-convex, while if X* has the Radon-Nikodym property, every compact operator is quasi-convex. Thus if K is a compact convex set such that the Banach space Ab(K) of all bounded affine functions on K has the Radon-Nikodym property, K is strongly locally convex (in the terminology of (9)), i.e. affinely homeomorphic to a subset of a locally convex space. We are also able to relate quasi-convexity of the range of a vector measure to the existence of a control measure. In 1947, Maharam (13) asked whether a sequentially continuous submeasure on a tr-algebra is equivalent to a measure. Applying the above results we relate Maharam's problem to the question of quasi-convexity of exhaustive operators on spaces C(ft) (see (11)). The author would like to thank J. Diestel, L. Drewnowski and Z. Lipecki for several helpful comments on a preliminary version of this note.