习题 13, § 5

令$E$为可数第一公理的非自反Banach 空间满足，并令$M$为无限余维$E$的非自反闭向量子空间（练习12）。 令$(x_n)$为单位球面上的处处稠密序列，$C$序列$(x_n/n)$的闭凸平衡包络线； $C$ 在$E$ 中是强紧致的，$C + M = A$ 是闭凸集合（GT，III, § 4, No. 1，cor.1 到 prop.1）。 Let $S$ be the unit ball in $E$, and $B = A \cap S$. a) Show that 0 is not an interior point of $A$ and deduce that there exists $x_0 \in E$ such that $\lambda x_0 \notin A$ for $\lambda > 0$. b) Show that there exists no supporting hyperplane of $B$ passing through 0 (observe that such a hyperplane must be a supporting hyperplane of $C$). c) Let $U_0 = M \cap S$, and let $(U_n)_{n \geq 1}$ be a decreasing sequence of closed convex, bounded and non-empty sets, such that $U_1 \subset \frac{1}{2} U_0$ and such that the intersection of the $U_n$ is empty (IV, p. 68, exerc. 8). 令 $F$ 为集合 $\frac{1}{n} x_0 + U_n$ ($n \geq 1$) 的并的闭凸包络。 证明$B \cap F = \varnothing$，但不存在分隔$B$ 和$F$ 的闭合超平面（使用$b$）。