Exercise 9, § 3

a) Let a, b be two points of a normed space E on the field R. Denote by $\delta(A)$ the diameter of a bounded set A in E (using the norm metric on E) and define inductively the sequence $(B_n)_{n \geq 1}$ of bounded sets in E satisfying the following conditions : $B_1$ is the set of those $x \in E$ such that $\|x - a\| = \|x - b\| = \frac{1}{2}\|a - b\|$; for $n > 1$, $B_n$ is the set of those $x \in B_{n-1}$ such that $\|x - y\| \leq \frac{1}{2}\delta(B_{n-1})$ for all $y \in B_{n-1}$. Show that the intersection of the $B_n$ is just the single point $\frac{1}{2}(a + b)$ (note that $\delta(B_n) \leq \frac{1}{2}\delta(B_{n-1})$). b) Deduce from a) that if $u$ is an isometry of the real Banach space E on the real Banach space F, then $u$ is an affine linear mapping of E on F.