Exercise 22, § 1

Let $E$ be a real normed space of dimension $\geqslant 3$ such that, for every homogeneous plane $P$ in $E$, there exists a continuous projector of $E$ onto $P$, of norm 1. Show that the norm on $E$ is prehilbertian. We shall reduce, with the aid of V, p. 59, exerc. 2, to the case where $E$ is of dimension 3, and we shall establish successively the following propositions: a) For every homogeneous plane $P$ in $E$, there exists a unique continuous projector of $E$ onto $P$, of norm 1, and the kernel of this projector is a homogeneous right $D(P)$, such that $P \mapsto D(P)$ is a continuous bijection of the space of homogeneous planes of $E$ onto the space of homogeneous rights of $E$ (TG, VI, p. 18). b) Every point of the sphere $S$ with equation $\|x\| = 1$ in $E$ is extremal in the ball $B$ of $E$ defined by $\|x\| \leqslant 1$. (Show first that, if $x \in S$ were not extremal, its facet $F_x$ in $B$ (II, p. 92, exerc. 3) would be of dimension 2, by considering all the homogeneous planes $P$ passing through $x$; prove next that this hypothesis is contradictory, proceeding in the same way at a point of $F_x$ where there exists only one supporting right of $F_x$ in the plane generated by $F_x$; the existence of such a point may be established by using II, p. 93, exerc. 7 and p. 94, 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 $\|x'\| \leqslant 1$. (Remark first that, for every homogeneous right $D'$ of $E'$, there exists one and only one homogeneous plane $P'(D')$ in $E'$ such that, for every point of $S' \cap P'(D')$, the supporting plane of $B'$ at this point (unique by a)) is parallel to $D'$; moreover the mapping $D' \mapsto P'(D')$ is continuous. Deduce from this first of all that if $x' \in S'$ were not extremal in $B'$, its facet $F_{x'}$ in $B'$ would be of dimension at least 2, by considering all the homogeneous rights $D'$ parallel to the supporting plane of $B'$ at the point $x'$. Show next that this hypothesis leads to a contradiction, by considering a point of strict convexity $y'$ of $F_{x'}$ (II, p. 94, exerc. 8), the unique homogeneous right $D'_0$ parallel to the supporting right of $F_{x'}$ at the point $y'$ in the plane generated by $F_{x'}$, and by proving 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 a same right $\Delta$, the three rights $D(P_1), D(P_2), D(P_3)$ are in the same homogeneous plane $\pi(\Delta)$ (consider the unique supporting plane of $B$ at a point of intersection of $\Delta$ and $S$). Applying the fundamental theorem of projective geometry (A, II, p. 203, exerc. 16), deduce that there exists a bijective linear mapping $\varphi$ of $E'$ onto $E$ such that, for every $x' \in E'$, the point $\varphi(x')$ belongs to the right $D(P)$, where $P$ is the plane with equation $\langle y, x' \rangle = 0$. Show that, for every point $x' \in S'$, the plane with equation $\langle \varphi(x'), y' - x' \rangle = 0$ is a supporting plane of $B'$ at this point, and conclude by applying V, p. 64, exerc. 20.