Exercise 9, § 4

Let E be a Hilbert space, $(P_j)_{j\in\mathbf{N}}$ a sequence of orthogonal projections in E, such that, for every $j\in\mathbf{N}$, there exists an $n_j\in\mathbf{N}$ such that for every $k\in\mathbf{N}$, at least one of the orthogonal projections $P_k,P_{k+1},\ldots,P_{k+n_j}$ is equal to $P_j$. We put $R_s=P_sP_{s-1}\ldots P_0$ for every $s\in\mathbf{N}$. (a) For every $x\in E$, put $x_s=R_s.x$. Prove 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 when $s$ tends to $+\infty$. (b) Let $(x_{s_k})$ be a subsequence of $(x_s)$ which tends weakly to a limit $y$; each sequence $(x_{s_k+r_k})$ then also tends weakly to $y$. 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 $s_k+r_k=j$; prove that the sequence $(x_{s_k+r_k})$ tends weakly to $y$.) (c) Prove that the sequence $(x_s)$ converges weakly to the orthogonal projection of $x$ 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.)