Exercise 15, § 3

Let E be an infinite-dimensional Banach space, $(a_n)_{n \geq 1}$ a countable total free family of points of E, $F_n$ the subspace of E of dimension $n$ generated by the $a_m$ of index $m \leq n$. Let $S_n$ be the sphere of equation $\|x\| = n$ in E; in $S_n \cap F_n$ let $A_n$ be a finite set such that every point of $S_n \cap F_n$ is at a distance $\leq 1/n$ from $A_n$. Now prove that $A = \bigcup_{n=1}^\infty A_n$ is such that its intersection with every closed bounded set is closed, but that 0 is adherent to A for the weak topology.