Corollary

— Let P be a pointed convex cone, with at least one interior point, of the topological vector space E. Then each linear form f that is not identically zero on E, and is positive for the preorder structure defined by P (II, p. 13), is necessarily continuous. Further, if x is interior to P then $f(x) > 0$ and if x is a cluster point of P then $f(x) \geqslant 0$.

Apply prop. 17 to the case $A = P$ where H is the hyperplane with the equation $f(x) = 0$.