Prove that for every orthogonal family of projectors $(P_n)$ on Hilbert space $H$ $$\sum\limits_{n=1}^{\infty} ||P_nf||^2 \leq ||f||^2, \forall f \in H.$$
From it, prove Bessel inequality $$\sum\limits_{n=1}^{\infty} |<f,e_n>|^2 \leq ||f||^2, \forall f \in H,$$
where $(e_n)$ is orthonormal system in $H$.
Orthogonal projector is $P \in B(H)$, such that $P^2=P=P^{*}$, and two orthogonal projectors $P$ and $Q$ are orthogonal if $PQ=0$.
I have proved Bessel if this is true – for arbitrary orthonormal system $(e_n)$ I took $P_n(f)=<f,e_n>e_n$, and I get $||P_n(f)||^2 = <P_nf, P_nf> = |<f,e_n>|^2$. But I don't know how to prove the first inequality.
Thanks!
Best Answer
Let $F_n:=P_n(H)$ so that $P_n$ is the projection onto $F_n$, and $N\geqslant 1$. The fact that the $P_n$ are orthogonal implies that we have an orthogonal direct sum $F_1\oplus\cdots\oplus F_N$, denote $F$ the latter, then $P_F:=P_1+\cdots+P_N$ is the projection onto $F$ so that $$ \sum_{n=1}^N\|P_nf\|^2=\|P_Ff\|^2=\|f\|^2-\|f-P_Ff\|^2\leqslant \|f\|^2 $$ by Pythagorean theorem. Letting $N\longrightarrow +\infty$ gives the desired inequality.