Exercise 9, § 4

Let E be a hilbertian space and $(P_j)_{j \in \mathbf{N}}$ be a sequence of orthoprojectors an E, such that, for all $j \in \mathbf{N}$, there exists an $n_j \in \mathbf{N}$ such that for all $k \in \mathbf{N}$, at least one of the orthoprojectors $P_k, P_{k+1}, ..., P_{k+n_j}$ is equal to $P_j$. Put $R_s = P_sP_{s-1} ... P_0$ for all $s \in \mathbf{N}$. a) For every $x \in E$, let $x_s = R_s.x$. Show that $\sum_s \|x_{s-1} - x_s\|^2 \leq \|x\|^2$, and deduce that for every integer $r \geq 1$, $x_{s+r} - x_s$ tends to 0 as s tends to $+\infty$. b) Let $(x_{s_k})$ be a sequence extracted from $(x_s)$ which tends to a limit $y$ weakly; then every sequence $(x_{s_k+r_k})$ also tends to $y$ weakly. Deduce that $y$ belongs to each of the subspaces $M_j = P_j(E)$. (For each $j$, there exists $r_k$ such that $0 \leq r_k \leq n_j$ and $s_k + r_k = j$; show that the sequence $(x_{s_k+r_k})$ tends to $y$ weakly.) c) Show that the sequence $(x_s)$ converges weakly to the orthogonal projection from x onto the intersection M of the $M_j$. (Reduce to the case where $M = \{0\}$, and use b) and the weak compactness of every closed ball in E.)