Exercise 6, § 4

Let $E$ be a Hilbert space, $M$, $N$ two closed subspaces of $E$, $M^{\circ}$, $N^{\circ}$ their respective orthogonal complements; denote by $p_M$, $p_N$ the orthogonal projections on M and N respectively, so 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 $\delta(M, N) = \|u_{NM}\|$; one has $\delta(M, N) = \delta(N^{\circ}, M^{\circ}) \leq 1$; the relation $\delta(M, N) < 1$ implies $M \cap N^{\circ} = \{0\}$. a) Denote by $\tilde{M}$ the orthogonal complement in M of $M \cap N^{\circ}$, and put $\varepsilon(M, N) = \delta(\tilde{M}, N)$. Now prove (with the notations of exerc. 5) that one has $\ell(u_{NM}) = M \cap N^{\circ}$ 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 $\delta(M, N) < 1$), one has $\varepsilon(M, N) = \delta(M, N)$. b) For a continuous linear mapping $u$ of E into itself, the conorm of $u$ is called the number $c(u) = \inf \|u(x)\|/\|x\|$, where $x$ runs through the set of vectors $\neq 0$ orthogonal to $u^{-1}(0)$ (if $u = 0$, put $c(u) = 1$). In order that $u(E)$ be closed in E, it is necessary and sufficient that $c(u) > 0$ (I, p. 17, th. 1). One has $c(u^*) = c(u)$. c) Put $v_{NM} = p_N p_M$. Now prove that one has

(note that $\|u_{NM}(x)\|^2 + \|v_{NM}(x)\|^2 = \|p_M.x\|^2$, 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. 63, exerc. 17). Now prove that one has (note 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 $M \cap N^{\circ} = N \cap M^{\circ} = \{0\}$, one has the relation $\varepsilon(M, N) = \delta(M, N) = \delta(N, M) = g(M, N)$. If $g(M, N) < 1$, one has $M \cap N^{\circ} = N \cap M^{\circ} = \{0\}$. f) Let $Q_M, Q_N$ be two continuous projections in E, having images M and N respectively; give another proof of the relation $g(M, N) \leq \|Q_M - Q_N\|$ (V, p. 63, exerc. 17). (Note that for every $x \in E$, one has $\|(1_E - Q_M).x\|^2 + \|Q_M^*.x\|^2 = \|x\|^2 + \|(Q_M - Q_M^*).x\|^2$, and apply this relation by taking $x = (p_M - p_N).y$ and noting that one has the relations $(1_E - Q_M)(p_M - p_N) = (Q_M - Q_N)p_N$ and $(p_M - p_N)Q_M = (1_E - p_N)(Q_M - Q_N)$.