Exercise 3, § 2

Let $E$ be a locally convex space, $L$ a vector subspace of $E$ of finite codimension, and $S$ a convex balanced bornivorous set (III, p. 39, exerc. 11) in $L$. We shall prove that there exists a convex, balanced and bornivorous set $S'$ in $E$ such that $S = S' \cap L$. a) We can reduce to the case where $L$ is a hyperplane such that $E = L \oplus \mathbf{R} a$ for a point $a \notin L$, and such that there exists a bounded sequence $(x_n)$ in $E$ such that if we put $x_n = \lambda_n (y_n + a)$ with $\lambda_n \in \mathbf{R}$ and $y_n \in L$, then $|\lambda_n|$ tends to $+\infty$; if $B_0$ is the convex balanced envelope of the set consisting of $a$ and the $x_n$, then we have $y_n + a \in \lambda_n^{-1} B_0$ for all $n$. b) Let $\mathscr{B}$ be the set of all bounded, convex, balanced subsets of $E$ which contain $B_0$; by hypothesis, for every $B \in \mathscr{B}$, there exists $\rho_B > 0$ such that $2\rho_B B \cap L \subset S$. Show that if $R$ is the convex balanced envelope of the union of the sets $\rho_B B$ for $B \in \mathscr{B}$, then we have** $R \cap L \subset S$.