Remark 1

Let C be a pointed proper convex cone in E, and consider on E the order structure for which C is the set of elements ≥ 0 (II, p. 12, prop. 13); in order that an element of E, say x > 0, belongs to an extremal generator of C, it is necessary and sufficient that every element y ≥ 0, that is bounded above by x, is of the form λx with 0 ≤ λ ≤ 1 : in fact, to say that y is bounded above by x means that x = y + y' where y' ∈ C, whence the conclusion follows.