If we have an orthonormal family, $\{u_n\}_{i=1}^\infty$ in a Hilbert Space $H$, I need to show that for $x\in H$ we have the following inequality:
$\left|\left\{n|\langle x, u_n \rangle > \frac{1}{m}\right\}\right|\leqslant m^2||x||^2.$
The only thing that seems relavent to me is the Bessel inequality $\sum_{n=1}^{\infty} |\langle mx,u_n\rangle |\leq ||mx||^2$ but this leads to trying to show that $\sum_{n=1}^{\infty} |\langle mx,u_n\rangle \geq |\{n||\langle x, u_n \rangle > \frac{1}{m}\}|$ which doesn't really help me.
It would be great if somebody could give me a hint as to how to go about this.
Thanks very much for any help.