Exercise 17, § 1

a) Let E be a real Hilbert space of dimension 2, $P_1$, $P_2$ two orthoprojectors in E, onto the lines $D_1$, $D_2$ respectively, supposed distinct. Prove that for every $x \in E$ such that $\|x\| = 1$, one has $\|(P_1 - P_2).x\| = \sin \theta$, where $\theta$ is the angle of $D_1$ and $D_2$ comprised between 0 and $\pi/2$, and that for every $y \neq 0$ in E, there exists $x \neq 0$ such that $(P_1 - P_2).x$ is collinear with $y$. b) Let E be a real Hilbert space, $P_1$, $P_2$ two orthoprojectors in E, with respective images $E_1$, $E_2$. Prove that $\|P_1 - P_2\|$ is the greatest lower bound of the numbers $\sin \theta$, where $\theta$ is the angle comprised between 0 and $\pi/2$ of two lines $D_1$, $D_2$ such that $D_1 \subset E_1$, $D_2 \subset E_2$, $D_1$ and $D_2$ being orthogonal to $E_1 \cap E_2$. c) Let $Q_1$, $Q_2$ be two continuous projectors in E, with images $E_1$, $E_2$, and let $P_1$, $P_2$ be the orthoprojectors onto $E_1$ and $E_2$ respectively. Prove that one has $\|P_2 - P_1\| \leq \|Q_2 - Q_1\|$. (Observe that one has $(Q_2 - Q_1)P_2 = (I - Q_1)(P_2 - P_1)$ and use a) and b).)