Exercise 6, § 4

Let $E$ be a hilbertian space, $M, N$ two closed subspaces of $E$ and $M^{\circ}, N^{\circ}$ their respective orthogonal complements; let $p_M, p_N$ denote the orthogonal projections on $M$ and $N$ respectively, such that $1_E - p_M, 1_E - p_N$ are the orthogonal projections on $M^{\circ}$ and $N^{\circ}$ respectively. Put $u_{NM} = (1_E - p_N) p_M$ and $\delta(M, N) = \|u_{NM}\|$; we have $\delta(M, N) = \delta(N^{\circ}, M^{\circ}) \leq 1$; the relation $\delta(M, N) < 1$ implies $M \cap N^{\circ} = \{0\}$. a) Let $\tilde{M}$ denote the orthogonal complement in $M$ of $M \cap N^{\circ}$, and let $\varepsilon(M, N) = \delta(\tilde{M}, N)$. Show that (with the notations of exerc. 5) $\ell(u_{NM}) = M \cap N^{\circ}$ and deduce that $\varepsilon(M, N) = \langle u_{NM} \rangle \leq \delta(M, N)$; moreover, if $M \cap N^{\circ} = \{0\}$ (and in particular if $\delta(M, N) < 1$), then $\varepsilon(M, N) = \delta(M, N)$. b) For a continuous linear mapping $u$ from $E$ into itself, we designate by conorm of $u$ the number $c(u) = \inf \|u(x)\| / \|x\|$, where $x$ ranges over the set of all vectors $\neq 0$ orthogonal to $u^{-1}(0)$ (if $u = 0$, put $c(u) = 1$). For $u(E)$ to be closed in $E$, it is necessary and sufficient that $c(u) > 0$ (I, p. 17, th. 1). We have $c(u^*) = c(u)$. c) Let $v_{NM} = p_N p_M$. Show that

(observe that $\|u_{NM}(x)\|^2 + \|v_{NM}(x)\|^2 = \|p_M \cdot x\|^2$, and that the kernel of $v_{NM}$ is $M^{\circ} + (M \cap N^{\circ})$ and deduce that $\langle u_{NM} \rangle^2 \leq 1 - c(v_{NM})^2$). d) Deduce from b) and c) that $\varepsilon(N, M) = \varepsilon(M, N)$ and, using a), that $\varepsilon(M^{\circ}, N^{\circ}) = \varepsilon(M, N)$. e) Put $g(M, N) = \|p_M - p_N\|$ (cf. V, p. 64, exerc. 17). Show that (observe that $p_M - p_N = (1_E - p_N) p_M - p_N (1_E - p_M)$); deduce that $\varepsilon(M, N) \leq g(M, N)$. If $M \cap N^{\circ} = N \cap M^{\circ} = \{0\}$, we have the relation $\varepsilon(M, N) = \delta(M, N) = \delta(N, M) = g(M, N)$. If $g(M, N) < 1$, then $M \cap N^{\circ} = N \cap M^{\circ} = \{0\}$. f) Let $Q_M, Q_N$ be two continuous projections in $E$, with respective images $M$ and $N$; give another proof of the relation $g(M, N) \leq \|Q_M - Q_N\|$ (V, p. 64, exerc. 17). (Note that for all $x \in E$, we have $\|(1_E - Q_M) \cdot x\|^2 + \|Q_M^* \cdot x\|^2 = \|x\|^2 + \|(Q_M - Q_M^*) \cdot x\|^2$, and apply this relation to $x = (p_M - p_N) \cdot y$, observing the relations $(1_E - Q_M) (p_M - p_N) = (Q_M - Q_N) p_N$ and $(p_M - p_N) Q_M = (1_E - p_N) (Q_M - Q_N)$.)