Exercise 14, § 2

Let A be a non-empty closed convex subset of E, a Hausdorff topological vector space. a) Show that, for every $a \in A$, the set $\bigcap_{\lambda > 0} \lambda(A - a)$ is a closed convex cone in E, with vertex 0, independent of $a$. It is called the asymptotic cone of A and written $C_A$. For every $a \in A$, the set $a + C_A$ is the union of $\{a\}$ and those open half lines that are contained in A and have $a$ as an end point. b) If $x, y$ are two points of A such that $(x + C_A) \cap (y + C_A)$ is a cone whose vertex $z \in A$, then this cone is necessarily $z + C_A$. c) If B is a second closed convex subset of E such that $A \cap B \neq \varnothing$, then $C_{A \cap B} = C_A \cap C_B$. d) Let $V_A$ be the largest vector subspace (necessarily closed in E) which is contained in $C_A$. Show that if $\phi$ is the canonical homomorphism of E on $E/V_A$, then $A = \phi^{-1}(A_0)$, where $A_0$ is a closed convex set in $E/V_A$ which does not contain any straight line. e) In the Banach space $\mathscr{B}(\mathbf{N})$ of bounded mappings of $\mathbf{N}$ in $\mathbf{R}$ (I, p. 4) give an example of a closed convex non-bounded set A, for which $C_A = \{0\}$ and which is such that for every $b \neq 0$ in E, there exists an integer $k$ for which $(A + kb) \cap A = \varnothing$.