Exercise 17, § 1

a) Let E be a real hilbertian space of dimension 2, $P_1, P_2$ two orthoprojectors from E onto the lines $D_1, D_2$ respectively, assumed distinct. Show that for every $x \in E$ such that $\|x\| = 1$, we have $\|(\hat{P}_1 - \hat{P}_2).x\| = \sin \theta$, where $\theta$ is the angle between $D_1$ and $D_2$ lying 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 hilbertian space, $P_1, P_2$ two orthoprojectors on E, with respective images $E_1, E_2$. Show that $\|P_1 - P_2\|$ is the lower bound of the numbers $\sin \theta$, where $\theta$ is the angle, between 0 and $\pi/2$ of the 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 on E, with images $E_1, E_2$, and let $P_1, P_2$ be the orthoprojectors onto $E_1$ and $E_2$ respectively. Show that $\|P_2 - P_1\| \leq \|Q_2 - Q_1\|$. (Observe that $(Q_2 - Q_1)P_2 = (I - Q_1)(P_2 - P_1)$ and use a) and b).)