Exercise 24, § 1

In a real normed space E of dimension $\geqslant 3$, we say that a vector $y$ is quasi-normal to a vector $x$, if for every scalar $\lambda$, we have $\|x + \lambda y\| \geqslant \|x\|$. a) Show 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. 65, exerc. 22 is satisfied). b) Show that the same conclusion holds, if for every closed homogeneous hyperplane H in E, there exists a vector $\neq 0$ which is quasi-normal to all the vectors of H. (Same method, apply th. 2 of E, III, § 2, No. 4 to the continuous projectors of norm 1 from the vector subspaces containing P onto a homogeneous plane P, these projections being linearly ordered by the relation of extension.) c) Show 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. 65, exerc. 22 to the dual of E). d) Show that the same conclusion holds if, when $z$ is quasi-normal to $x$ and $y$, then $z$ is quasi-normal to $x + y$ (apply V, p. 65, exerc. 22).