Let E be an ordered locally
convex topological vector space whose positive cone is normal, closed and generating.
It is an important problem to characterize those spaces E, whose topological dual E′
is a lattice for the dual ordering. It is proved here, with the use of Choquet’s
theory of weakly complete cones, that if E is a Fréchet space, E′ is lattice
if and only if E has the Riesz decomposition property. In fact, a stronger
result is proved which is, even in the Banach case, an improvement of the
classical Andô’s theorem. An application to the duality of order ideals is
given.
