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I am trouble proving the following proposition in Conway's functional analysis book. $H$ is an arbitrary Hilbert space, $I$ is an index set.

Prop - Let $\{P_i:i\in I\}$ be a family of pairwise orthogonal projections in $B(H)$. That is $P_iP_j=0$ for $i\neq j$. If $h\in H$, then $\sum_i\{P_ih:i\in I\}$ converges in $H$ to $Ph$, where $P$ is the projection of $H$ onto closed linear span of $\{P_iH:i\in I\}$.

I am trying like this:

Proof - Let $M$ be the closed linear span of $\oplus_iP_iH$, then we can write $H$ as $H=\oplus_iM_i+K$, where of course $K$ is orthogonal complement of $M$. Then for $h\in H$, we can write $h=\sum_iP_ih+k=\sum_ih_i+k$. Hence applying $P$ both sides gives $Ph=\sum_iP_ih$, since $Pk=0$.

But I am not entirely convinced by this approach, and I feel like I am oversimplifying things. Am I correct, or what am I overlooking?

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    You give a definition of convergence, but you are not using this definition (or stating a criterion for convergence) to prove convergence. So your proof cannot be right.2012-06-13

2 Answers 2

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Let $M_j=P_jH$, $j\in I$. Let $K=\left(\cup_jM_j\right)^\perp$.

Claim. For any $h\in H$, there exists a unique net $(h_j)_j$, with $h_j\in M_j$, and $k\in K$, also unique, such that $\tag{1} h=k+\lim_F \sum_{j\in F}h_j. $

Assuming the claim, and taking into account that the limit in $(1)$ is in norm, and that $P_i$ and $P$ are continuous, we have, for any $i\in I$, $ P_ih=P_ik+\lim_F\sum_{j\in F}P_ih_j=h_i\,; $ and $ Ph=Pk+\lim_F\sum_{j\in F}Ph_j=\lim_F\sum_{j\in F}h_j=\lim_F\sum_{j\in F}P_jh $

So it remains to prove the claim. Note that $M_i\perp M_j$ if $i\ne j$. Let $M$ be the subspace of $H$ generated by $\cup_jM_j$. For each $j\in I$, let $\{f_{j,n}\}_{n\in I_j}$ be an orthonormal basis of $M_j$. Then $\{f_{j,n}\}_{j\in I, n\in I_j}$ is an orthonormal basis for $M$ (easy to check).

Let $h\in H$. If we write $k$ for the projection of $h$ onto $M^\perp$, we have $h-k\in M$. So $ h-k=\sum_{j\in I, n\in I_j} \alpha_{j,n}\;f_{j,n}. $ This means that for any $\varepsilon>0$ there exists a finite set $F\subset \{(j,n):\ j\in I, n\in I_j\}$ such that if $F'\supset F$ then $\|h-k-\sum_{F'}\alpha_{j,n}\,f_{j,n}\|<\varepsilon$. Using that the net $\{\alpha_{j,n}\}$ is in $\ell^2$, we can define $ h_j=\sum_{n\in I_j}\alpha_{j,n}\;f_{j,n},\ \ \ j\in I. $ So $h-k=\sum_{j\in I} h_j$. So $h=k+\sum_{j\in I}h_j$. The uniqueness is straightforward using orthogonality.

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    Thank you, Prof. Martin. I also managed to work this out; though I showed that $\sum_ih_i$ is a Cauchy net, and by completeness it will converge (to $h-k$). I am feeling more confident in using nets.2012-06-17
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Well !!! (thank you !) I think that $h \in \oplus_i P_i H \oplus K $ means only that $h= \oplus P_i x_i + k $ for some $(x_i) \in H^I$

Isn't it ?

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    More explanation : when you say $y \in P_1H \oplus P_2 H$ , that means $y=P_1(x_1) + P_2(x_2)$ for some $x_1 \in H$ and some $x_2 \in H$.2012-06-13