Exercise 29, § 7

a) Let $(E_n)$ be a sequence of Hausdorff locally convex spaces and $E = \prod E_n$ their product. In each $E_n$, let $C_n$ be a convex cone with vertex 0 and $A_n$ a cap of $C_n$. Show that there exists a cap of $C = \prod C_n$ which contains $\prod A_n$ (argue as in prop. 5 of II, p. 59).

b) Let $(E_n, \phi_{nm})$ be an enumerable directed projective system of Hausdorff locally convex spaces and let $E = \lim_{\leftarrow n} E_n$ be its projective limit. For all $n$, let $C_n$ be a convex cone of vertex 0 such that $(C_n)$ is a projective system of sets. Show that if, for each $n$, the set $C_n$ is the union of its caps, then this is also true of $C = \lim_{\leftarrow n} C_n$ (use $a$). In particular, if the $C_n$ are cones with compact soles then $C$ in the closed convex envelope of the union of its extremal generators.