If the weak limit of sequence of projections is again a projection then strong limit exists.

Question

Let $\{P_n\}_{n \geq 1}$ be a sequence of orthogonal projections on a Hilbert space $\mathcal H$ such that weak limit of the sequence is again a projection $P$ i.e. $\left \langle x, P_n x \right \rangle \rightarrow \left \langle x, P x \right \rangle.$ Show that $P_n \rightarrow P$ in strong operator topology i.e. $P_n x \rightarrow P x.$

What we have given is that $\|P_n x\|^2 \rightarrow \|Px\|^2.$ But I don't understand how does it follow the result. Any help in this regard will be warmly appreciated.

Thanks for reading.

Answer 1

By using polarization, $\langle y,P_nx\rangle\to\langle y,Px\rangle$ for all $x,y$. Then you have \begin{align} \|P_nx-Px\|^2 &=\langle P_nx-Px,P_nx-Px\rangle\\[0.3cm] &=\langle P_nx,P_nx\rangle+\langle Px,Px\rangle-2\operatorname{Re} \langle Px,P_nx\rangle\\[0.3cm] &=\langle x,P_nx\rangle+\langle x,Px\rangle-2\operatorname{Re} \langle Px,P_nx\rangle\\[0.3cm] &\to2\langle x,Px\rangle-2\operatorname{Re}\langle Px,Px\rangle =2\langle x,Px\rangle-2\langle x,Px\rangle\\[0.3cm] &=0 \end{align}