Question: 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?

Answer: 
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.
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 ?