Exercise 36, § 7

Consider the space E = R^{(N)}, with the weak topology σ(R^{(N)}, R^N); let C be the closed convex cone in E formed by the points x = (x_n) such that x_n ≥ 0 for all n. a) Let x = (x_n) be a point of C and let J be the finite set of integers n for which x_n > 0; if m is the number of elements of J, then let A be the set of the points y = (y_n) of C such that y_n = 0 for n ∈ J and $\sum_{k\in J} y_k x_k^{-1} \leq m$. Show that A is a cap of C containing x. b) Show that there does not exist a cap B in C such that C is the union of the sets nB for n > 0. (Let p be the restriction of the gauge of B to C; p will be finite in C and, if (e_n) is the canonical basis of E, we have p(e_n) > 0 for all n (II, p. 58, prop. 4), and the points z^{(n)} = e_n/p(e_n) belong to B; but show that there exists z' ∈ R^N such that the sequence (⟨z^{(n)}, z'⟩) is not bounded.)