Exercise 23, § 1

Let E be a complex normed vector space of dimension $\geqslant 3$, such that for every (complex) homogeneous plane P in E, there exists a continuous projector from E onto P with norm 1. Show that the norm on E is prehilbertian. Using V, p. 60, exerc. 2 reduce to the case where E is of dimension 3 over $\mathbf{C}$, and proceed as in exerc. 22. (For part b) of the proof, consider, for every $x' \in E'$ such that $\|x'\| = 1$, the convex set $G_{x'}$ of all $x \in S$ such that $\langle x, x' \rangle = 1$, show that if $G_{x'}$ is not simply 0, it would have dimension at least 3 over $\mathbf{R}$; then in the real affine linear variety generated by $G_{x'}$, consider a boundary point of $G_{x'}$, where there exists only one support hyperplane (real) of $G_{x'}$. Similarly, for part c) of the proof, consider, for all $x \in S$, the set $G'_x$ of all $x' \in S'$ such that $\langle x, x' \rangle = 1$ and show that $G'_x$ reduces to a point; for this, prove that, if not, the real affine linear variety generated by $G'_x$ will have dimension at least 2 over $\mathbf{R}$, and will contain two linearly independent vectors over $\mathbf{C}$. Conclude using V, p. 65, exerc. 21.)