Exercise 22, § 1

Let $E$ be a real normed space of dimension $\geq 3$, such that, for every homogeneous plane $P$ in $E$, there exists a continuous projector from $E$ onto $P$, of norm 1. Show that the norm on $E$ is prehilbertian. With the help of V, p. 60, exerc. 2, reduce to the case where $E$ is of dimension 3, and establish successively the following propositions. a) For every homogeneous plane $P$ in $E$, there exists a unique continuous projector from $E$ onto $P$, with norm 1, and the kernel of this projection is a homogeneous line $D(P)$ such that $P \mapsto D(P)$ is a continuous bijection from the space of homogeneous planes of $E$ into the space of homogeneous lines of $E$ (GT, VI, § 3, No. 5). b) Every point of the sphere $D : \|x\| = 1$ in $E$ is extremal in the ball $B$ of $E$ defined by $\|x\| \leq 1$. (First show that if $x \in S$ is not extremal, its section $F_x$ in $B$ (II, p. 87, exerc. 3) will be 2-dimensional, by considering all the homogeneous planes $\tilde{P}$ passing through $x$, next prove that this hypothesis is contradictory, by proceeding similarly at a point in $F_x$ where there exists only one support line of $F_x$ in the plane generated by $F_x$; the existence of such a point can be established by using II, p. 88, exerc. 7 and p. 88, exerc. 8). c) Every point of the sphere $S'$ with equation $\|x'\| = 1$ in the dual $E'$ of $E$ is extremal in the ball $B'$ of $E'$ defined by $\|x'\| \leq 1$. (First observe that for every homogeneous line $D'$ of $E'$, there exists a unique homogeneous plane $P'(D')$ in $E'$ such that for every point in $S' \cap P'(D')$, the support plane of B' at this point (unique by $a$) is parallel to D'; moreover, the mapping $D' \mapsto P'(D')$ is continuous. Deduce that if $x' \in S'$ were not an extremal point in B', its section $F_{x'}$ in B' would be of dimension at least 2; for this consider all the homogeneous lines D' parallel to the support plane of B' at the point $x'$. Next show that this hypothesis implies a contradiction, by considering a point of strict convexity $y'$ of $F_{x'}$ (II, p. 88, exerc. 8), and the unique homogeneous line $D'_0$ parallel to the support line of $F_{x'}$ at the point $y'$ in the plane generated by $F_{x'}$, and prove that the function $D' \mapsto P'(D')$ would not be continuous for $D' = D'_0$. d) Show that, if three homogeneous planes $P_1, P_2, P_3$ in E contain the same line $\Delta$, then the three lines $D(P_1), D(P_2), D(P_3)$ are in the same homogeneous plane $\pi(\Delta)$ (consider the unique support plane of B at a point of intersection of $\Delta$ and of S). Applying the fundamental theorem of projective geometry (A, II, § 9, exerc. 16) deduce that there exists a bijective linear mapping $\phi$ from E' onto E such that, for all $x' \in E'$, the point $\phi(x')$ belongs to the line D(P), where P is the plane with the equation $\langle y, x' \rangle = 0$. Show that for every point $x' \in S'$, the plane with the equation $\langle \phi(x'), y' - x' \rangle = 0$ is the support plane of B' at this point, and conclude by applying V, p. 65, exerc. 20.