Exercise 4, § 5

Let $E$ be a normed space and $A$ a complete convex set for the uniform structure induced by that of $E$. a) Let $x'$ be a continuous linear form on $E$ that is bounded in $A$. Consider a number $k > 0$ and the closed convex cone $P$ in $E$, with vertex 0 and formed by the $x \in E$ such that $\|x\| \leq k \langle x, x' \rangle$; it is pointed and proper. Show that for the order on $E$ for which $P$ is the set of elements $\geq 0$, the set $A$ is inductive (use the fact that the restriction of $x'$ to $A$ is increasing and bounded). b) Deduce from a) that the set of points of the frontier $F$ of $A$ which belong to a support hyperplane of $A$ is dense in $F$ (Bishop-Phelps th.). (For each point $z \in F$, consider a point $y \in \mathscr{C} A$ arbitrarily close to $z$, and separate $y$ strictly from $A$ by a closed hyperplane of equation $\langle x, x' \rangle = \alpha$, with $\|x'\| = 1$ and use a) with $k > 1$, also exerc. 3 above.)