Exercise 33, § 7

Let $E$ be a Hausdorff locally convex space and $A$ a subset of $E$. a) Denote by $\Gamma_0(A)$ the set of points $x \in E$ such that, for every continuous linear mapping $u$ of $E$ in a finite dimensional vector space, the image $u(x)$ belongs to the convex envelope of $u(A)$. This comes to the same as saying that for every closed linear variety $V$ of $E$ containing $x$ and of finite codimension $n > 0$, there exists a subset of $A$ having at most $n + 1$ elements, and of which the convex envelope meets $V$. Show that $\Gamma_0(A)$ is a convex set containing $A$, that $\Gamma_0(\Gamma_0(A)) = \Gamma_0(A)$ and that $\Gamma_0(A)$ is contained in the closed convex envelope of $A$ (use prop. 4 of II, p. 38). b) Let $(x_\alpha)_{\alpha \in I}$ be a family of elements of $A$ and $(\lambda_\alpha)_{\alpha \in I}$ a family of positive numbers such that $\sum_{\alpha \in I} \lambda_\alpha = 1$ and that the family $(\lambda_\alpha x_\alpha)$ is summable in $E$. Show that the sum $s = \sum_{\alpha \in I} \lambda_\alpha x_\alpha$ belongs to $\Gamma_0(A)$. (With the aid of $a$) reduce to the case where $E$ is of finite dimension and identical with the linear variety generated by the $\lambda_\alpha x_\alpha$; then argue by reductio ad absurdum, considering, for every finite subset $J$ of $I$, a closed hyperplane $H_J$ that passes through $s$ and does not meet the convex envelope of the set of the $x_\alpha$ such that $\alpha \in J$, then using the compactness of the unit sphere in a finite dimensional space.) c) Show that if $A$ is compact, then $\Gamma_0(A)$ is identical with the convex closed envelope of $A$. d) If $K$ is a compact convex set in $E$, and $A$ the set of its extremal points show that $K = \Gamma_0(A)$ (use exerc. 22, b) of II, p. 90, and exerc. 32). e) With the notations of II, p. 74, exerc. 3, let $A$ be the set formed of the $\varepsilon_x$, where $x$ varies in the set of rational numbers such that $0 \leq x \leq 1$. Show that $\Gamma_0(A)$ is distinct from the convex envelope of $A$ and from the convex closed envelope of $A$.

EXERCISES

TVS II.93