Let $\phi_1, \phi_2,\dots$ be a complete orthonormal system in a Hilbert space. Define vectors by
$\psi_n=C_n(\sum_{k=1}^n \phi_k-n\phi_{n+1})$
$(n=1, 2, \dots)$.
(i) Show that $\psi_1, \psi_2, \dots$ form an orthogonal system in this Hilbert space.
(ii) Find constants $C_n$ that make $\psi_1, \psi_2, \dots$ into an orthonormal system.
(iii) Show that if $(\psi_n, x)=0$ for all $n=1, 2, \dots$, then $x=0$.
(iv) Verify:
$||\phi_k||^2=1=\sum_{n=1}^{\infty} |(\psi_n, \phi_k)|^2$
I'm self-studying some mathematical physics and really quite lost on this problem. I have read the entire chapter on Hilbert spaces, but am still very lost as to where to begin. I'm looking for someone to help me with this problem. Thanks!