Exercise 5, § 1

Let E be a real or complex vector space, of dimension $\geq 2$. Suppose a mapping $x \mapsto \|x\|$ from E into $\mathbf{R}_+$ is given, satisfying : $\|\lambda x\| = |\lambda| \cdot \|x\|$ for every scalar $\lambda$, that $\|x\| = 0$ implies $x = 0$ and that we have the « ptolemaic inequality »

for every $a, b, c, d$ in E. a) Show that $\|x\|$ is a norm on E (replace $d$ by 0 and $b$ by $-a$).

b) Show that the norm on E is prehilbertian. (From the ptolemaic inequality deduce the inequality $\|x + y\|^2 + \|x - y\|^2 \geq 4 \|x\| \cdot \|y\|$ and use exerc. 4, c.) Prove the converse (show that in a hilbertian space, if we put $a' = a/\|a\|^2, b' = b/\|b\|^2$, we have the equality $\|a' - b'\| = \|a - b\|/\|a\| \cdot \|b\|$). If $\|a\| = \|b\| = \|c\| = \|d\|$ and if the four vectors $a, b, c, d$ are in the same plane, the two members of the ptolemaic inequality are equal.