Exercise 6, § 1

a) Let E be a real or complex normed space, of dimension $\geq 2$. Show that for every $x \neq 0$ in E and every real number $\alpha > 0$, there exists an element $y$ in E such that $\|y\| = \alpha$ and $\|x + y\|^2 = \|x\|^2 + \|y\|^2$. b) Suppose that if the vectors $x, y$ in E satisfy the relation $\|x + y\|^2 = \|x\|^2 + \|y\|^2$, then we also have $\|x - y\|^2 = \|x\|^2 + \|y\|^2$. Show that the norm on E is prehilbertian. (Using a), reduce to the case in exerc. 4 : restricting to the case where E is 2-bidimensional, we prove that if $\|x_1\| = \|x_2\| = 1, y = \frac{1}{2}(x_1 - x_2)$ and if $z \in E$ is such that $\|y\|^2 + \|z\|^2 = \|y + z\|^2 = 1$, then $z = \frac{1}{2}(x_1 + x_2)$ or $z = -\frac{1}{2}(x_1 + x_2)$. c) Suppose that, for every vector $x \neq 0$ in E, the set H of all vectors $y$ satisfying $\|x - y\|^2 = \|x\|^2 + \|y\|^2$ is stable under addition. Show that the conclusion of b) holds. (Reduce to the case where E is real and of dimension 2. Using a) and the compactness of the unit ball in E, show that H is a closed set containing at least two distinct half-lines with origin 0 ; prove that these two half lines are opposite to each other by observing that, if not, the convex set which they generate would be contained in H and will contain either $x$ or $-x$.)