Exercise 35, § 7

Let I be a non enumerable set, write E = R^{(I)} and E' = E × R. Denote the canonical basis of E by (e_α)_{α∈I} and let s be the element (0, 1) of E'. Define a separating duality between E and E', by ⟨e_α, e_β⟩ = δ_{αβ}, ⟨e_α, s⟩ = 1 for all α ∈ I. Let C be the pointed cone R^I_+ in E. a) Show that the topologies induced on C by σ(E, E') and by the norm p(x) = ∑_{α∈I} |x_α| on E coincide. b) Show that the uniform structure induced on C by σ(E, E') is not metrisable.