Exercise 15, § 1

In a Hausdorff topological vector space E over $\mathbf{R}$, let A be a compact convex set and B a closed, convex and bounded set. Show that the convex envelope C of the union $A \cup B$ is a closed set. (Consider a point z in the closure of C, but not in A, and reduce to the case where $z = 0$. Observe that there exists a neighbourhood V of 0 and a number $\alpha < 1$ such that the relations $0 \leq \lambda \leq 1,\ x \in A,\ y \in B,\ \lambda x + (1 - \lambda) y \in V$ imply that $\lambda \leq \alpha$. Next, for every neighbourhood W of 0 consider the set of triplets $(\lambda, x, y)$ such that $\lambda x + (1 - \lambda) y \in W,\ 0 \leq \lambda \leq 1,\ x \in A,\ y \in B.$)