Exercise 12, § 4

Let $E$ be a real Hilbert space, $V$ a closed convex cone in $E$, with vertex 0, $V^\circ$ the polar cone of $V$ (in $E$, canonically identified with its dual). a) Now prove that every point $x \in E$ can be written in a unique way in the form $x = x_+ - x_-$ where $x_+ \in V$ and $x_- \in V^\circ$, and $\langle x_+|x_- \rangle = 0$. b) For every face $F$ of $V$ (II, p. 92, exerc. 3), $F$ is reduced to 0 or is a convex cone with vertex 0; the set of the $y \in V^\circ$ which are orthogonal to $F$ is a closed face $F'$ of $V^\circ$ (but it is not the “dual face” of $F$ in the sense of II, p. 93, exerc. 6, the latter being empty). c) Take for $E$ the set of Hilbert-Schmidt endomorphisms of a real Hilbert space $H$, and for $V$ the set of positive elements in $E$. Now prove that one has $V^\circ = V$ and interpret in this case the result of a) (use cor. 1 of V, p. 56 to see that $V \subset V^\circ$). d) The hypotheses being those of c), let $v \in V$; the set $L$ of the $x \in H$ such that $\langle v(x)|x \rangle = 0$ or, equivalently, $v(x) = 0$ (V, p. 76, exerc. 10) is a closed vector subspace of $H$, and the face $F$ of $v$ in $V$ is the closed set of the $u \in V$ such that $u(x) = 0$ for all $x \in L$; it is identified with the cone of positive Hilbert-Schmidt endomorphisms of the Hilbert space $L^\circ$. Deduce that the projection of $E$ onto the convex set $F$ (V, p. 11) is identical with the orthogonal projector of $E$ onto the closed vector subspace of $E$ generated by $F$.