Regarding convergence of infinite sum of projections in norm

Question

Let $\{P_n\}$ be a sequence of pairwise orthogonal non- zero projections in $B(H)$. That is $\{P_nP_m=P_mP_m=0\; \text{for} \;n\neq m\}$. If $P=\sum_{n}P_n$ , then how do I show that $\|P-\sum_{j=1}^{n}P_n\|=1$ for all $n$?

Answer 1

I am assuming that $P_n$'s are non-zero orthogonal projections. Note that $P_n x=0$ whenever $x$ is in the range of $P_m$ with $n \neq m$. If $x$ is a unit vector in the range of $P_{n+1}$ then $Px=x$ and $\sum\limits_{k=1}^{n} P_k (x)=0$ so $\|P-\sum\limits_{k=1}^{n} P_k\| \geq 1$. Do you know how to prove the reverse inequality? I will add a proof if you cannot do this.

Hint: If $P$ and $Q$ are orthogonal projections with $PQ=QP=0$ then $P+Q$ is also a projection. Hence all finite sums of $P_i$'s are orthogonal projections.