Exercise 7, § 1

Let E be a real or complex normed space, with dimension $\geq 2$, having the following property : there exists a real number $\gamma$ distinct from 0 and from $\pm 1$, such that the relation $\|x + y\| = \|x - y\|$ implies $\|x + \gamma y\| = \|x - \gamma y\|$. a) Show that if $\|x + y\| = \|x - y\|$, the convex mapping $\phi : \xi \mapsto \|x + \xi y\|$ from $\mathbf{R}$ into $\mathbf{R}$ is not constant on any interval. (Argue by reductio ad absurdum ; let $[\alpha, \beta]$ be the largest interval on which $\phi$ is constant, show that there exists $\delta > \beta$ close enough to $\beta$ and such that $\phi(\delta) = \phi(\beta)$, by observing that if $\|u + v\| = \|u - v\|$, then $\|u + \gamma^n v\| = \|u - \gamma^n v\|$ for every rational integer $n$.) b) Show that if $\|x + y\| = \|x - y\|$, then $\|x + \xi y\| = \|x - \xi y\|$ for every real number $\xi$. (With the notations of a), observe that $\phi$ has a relative minimum at the point $\xi = 0$, using the fact that $\phi(\gamma^n) = \phi(-\gamma^n)$ for every rational integer $n$; deduce that we have $\phi(\xi) = \phi(-\xi)$ identically, for, otherwise, we get $\phi(\lambda) = \phi(\mu)$ for two numbers $\lambda, \mu$ such that $\lambda + \mu \neq 0$ and that in this case $\phi$ has a relative minimum at the point $\frac{1}{2}(\lambda + \mu)$.) c) Deduce from b) that the norm on E is prehilbertian (show first that if $\|x\| = \|y\|$, we have $\|\alpha x + \beta y\| = \|\beta x + 2y\|$ for every pair of real numbers $\alpha, \beta$ and that the equality $\|x + y\| = \|x - y\|$ implies $\|\alpha x + \beta y\| = \|\alpha x - \beta y\|$ for every pair of real numbers $\alpha, \beta$. Next, show that if $\|x\| = \|y\| = 1$ and $\|x + y\| = \|x - y\|$, we have the relation $\|(\alpha^2 - \beta^2)x + 2\alpha\beta y\| = \alpha^2 + \beta^2$ by using the preceding results, and deduce the conclusion).