Exercise 37, § 7

Let F be the Banach space l^1(N) of summable sequences x = (x_n) of real numbers and let E be the space of sequences y = (y_n) which tend to 0; we give F the weak topology σ(F, E) where E and F are in separating duality using the form B(x, y) = ∑ x_n y_n. a) Let C be the convex cone in F formed of the points x = (x_n) such that x_n ≥ 0 for all n. Show that C is closed in F. b) Let A be the set of the x = (x_n) ∈ C such that $\sum^n x_n \leq 1$. Show that A is a cap of C, that is metrisable for the topology induced by that of F, and that C is the union of the sets nA for n > 0. c) Show that a sole of C is not compact (such a set S would be the set of the x = (x_n) ∈ C such that $\sum^n z_n x_n = 1$, where (z_n) ∈ E and z_n > 0 for all n. If e_m = (δ_{mn})_{n≥0}, the points $z_n^{-1} e_n$ belong to S, but do not form a relatively compact set in F). d) Show that C is not metrisable for the topology induced by that of F (use Baire's th., noting that there is no point in A that is an interior point relative to the subspace C).