Prove limit properties associated with an ascending chain of closed Hilbert subspaces

Question

Please help with this problem, I have it in my class of econometrics but it is functional analysis:

Answer 1

1) Let $P_0=0$. Consider $(P_j-P_{j-1})X$ and $(P_i-P_{i-1})X$ for $j>i$. Then $(P_i-P_{i-1})X\in H_{j-1}$. Note that $P_{j-1}X=P_{j-1}P_jX$ since $H_{j-1}\subset H_j$, so $(P_j-P_{j-1})X=(I-P_{j-1})(P_jX)\perp H_{j-1}$.

2) 3): sorry I have no clue at his moment.

Answer 2

2) Write $P_nx = P_1x + \sum_{j=1}^{n-1} (P_{j+1}-P_j)x$, then using the orthogonality from (1) we get $$ \|P_nx \|^2 = \|P_1x\|^2 + \sum_{j=1}^{n-1} \|(P_{j+1}-P_j)x\|^2. $$ The left-hand side is bounded by $\|x\|^2$, hence the sum on the right hand side stays bounded for $n\to\infty$.

3) First prove that $(P_nx)$ is a Cauchy sequence: let $n>m$ be given then as in (2) we get $$ \|P_nx-P_mx\|^2 = \sum_{j=m}^{n-1} \|(P_{j+1}-P_j)x\|^2 \le \sum_{j=m}^\infty \|(P_{j+1}-P_j)x\|^2 . $$ The right hand side can be made arbitrarily small for large $m$. This proves $P_nx\to y$. Since $P_nx\in H_\infty$ for all $n$, it follows $y\in H_\infty$. It remains to shot that $x-y \in H_\infty^\perp$.

Since the linear hull of the union of the $H_n$'s is dense in $H_\infty$, it is enough to show that $x-y\in H_n^\perp$ for all $n$. Take $z\in H_m$. Then for $n>m$ we get $(x-P_nx,z)=0$ since $z\in H_n \supset H_m$. Passing to the limit yields $(x-y,z)=0$ for all $z\in H_n$ for all $n$. By density, this holds for all $z\in H_\infty$.