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?