Exercise 23, § 1

Let $E$ be a complex normed vector space of dimension $\geqslant 3$, such that for every homogeneous (complex) plane $P$ in $E$, there exists a continuous projector of $E$ onto $P$, of norm 1. Show that the norm on $E$ is pre-Hilbertian. We shall reduce, using V, p. 59, Exer. 2, to the case where $E$ is of dimension 3 over $\mathbf{C}$, and we shall proceed as in Exer. 22. (For part b) of the proof, consider, for every $x' \in E'$ such that $\|x'\| = 1$, the convex set $G_{x'}$ of the $x \in S$ such that $\langle x, x' \rangle = 1$; show that if $G_{x'}$ were not reduced to a point, it would be at least of dimension 3 over $\mathbf{R}$; consider then, in the real affine linear variety generated by $G_{x'}$, a boundary point of $G_{x'}$ where there exists only a single supporting (real) hyperplane of $G_{x'}$. For part c) of the proof, consider similarly, for every $x \in S$, the set $G'_x$ of the $x' \in S'$ such that $\langle x, x' \rangle = 1$, and show that $G'_x$ is reduced to a point; for this, prove that, in the contrary case, the real affine linear variety generated by $G'_x$ would be at least of dimension 2 over $\mathbf{R}$, and would contain two vectors linearly independent over $\mathbf{C}$. Conclude using V, p. 64, Exer. 21.)