Exercise 24, § 1

In a real normed space $E$ of dimension $\geqslant 3$, one says that a vector $y$ is quasi-normal to a vector $x$ if, for every scalar $\lambda$, one has $\|x + \lambda y\| \geqslant \|x\|$. a) Now prove that, if the relation « $y$ is quasi-normal to $x$ » is symmetric in $x, y$, then the norm on $E$ is prehilbertian. (Show that the condition of V, p. 64, Exercise 22 is satisfied.) b) Now prove that the same conclusion holds if, for every closed homogeneous hyperplane $H$ in $E$, there exists a vector $\neq 0$ quasi-normal to all the vectors of $H$. (Same method, applying Theorem 2 of E, III, p. 20 to the continuous projections of norm 1, on a homogeneous plane $P$, of vector subspaces containing $P$, these projections being ordered by the relation of prolongation.) c) Now prove that the same conclusion holds if, for every vector $x \neq 0$ in $E$, there exists a closed hyperplane $H$ such that $x$ is quasi-normal to all the vectors of $H$. (Reduce to the case where $E$ is of dimension 3, and apply V, p. 64, Exercise 22 to the dual of $E$.) d) Now prove that the same conclusion holds if, when $z$ is quasi-normal to $x$ and $y$, $z$ is quasi-normal to $x + y$ (apply V, p. 64, Exercise 22).