Exercise 12, § 4

Let $E$ be a real hilbertian space and $V$ be a closed convex cone in $E$, with vertex 0, let $V^\circ$ be the polar cone of $V$ (in $E$, identified canonically with its dual). a) Show that every point $x \in E$ can be written uniquely in the form $x = x_+ - x_-$ where $x_+ \in V$ and $x_- \in V^\circ$, and $\langle x_+|x_- \rangle = 0$. b) For every facet $F$ of $V$ (II, p. 87, exerc. 3), $F$ is either the point 0 or is a convex cone with vertex 0 ; the set of all $y \in V^\circ$ which are orthogonal to $F$ is a closed facet $F'$ of $V^\circ$ (but this is not the « dual facet » of $F$ in the sense of II, p. 87, exerc. 6, the latter being empty). c) Take for $E$ the set of all Hilbert-Schmidt endomorphisms of a real hilbertian space, and for $V$ the set of positive elements in $E$. Show that we have $V^\circ = V$ and in this case interpret the result of $a$ (to see that $V \subset V^\circ$, use cor. 1 of V, p. 56). d) Under the hypothesis of $c$, let $v \in V$; the set $L$ of all $x \in H$ such that $\langle v(x)|x \rangle = 0$, or, which is the same, such that $v(x) = 0$ (V, p. 77, exerc. 10) is a closed vector subspace of $H$, and the facet $F$ of $v$ in $V$ is the closed set of all $u \in V$ such that $u(x) = 0$ for all $x \in L$; it can be identified with the cone of all positive Hilbert-Schmidt endomorphisms of the hilbertian space $L^\circ$. Deduce that the projection from $E$ onto the convex set $F$ (V, p. 11) is identical with the orthogonal projector from $E$ onto the closed vector subspace of $E$ generated by $F$.