I'm trying to show that, given a spectral measure $d\mu_\psi(\lambda)$ for a self-adjont operator $A$, for the following quadratic form
$$q_\lambda(\psi)=\int_{\mathbb R}\chi_{(-\infty,\lambda]}(\tau) d\mu_\psi (\tau)$$
there exists an operator $P(\lambda)$ such that:
$$<\psi|P(\lambda)\psi>= q_\lambda(\psi)$$
In particular, $q_\lambda(\psi)$ is bounded from below and so I have just to prove that it is closed.
Well, I know that in order to show closedness I should prove that the domain of the quadratic form $Q$ is complete with respect to the norm form
$$||\psi||_q= q_\lambda(\psi)+||\psi||_H$$ where $H$ is our generic Hilbert space.
I have some troubles proving this, since chosen a Cauchy sequence on $H$, I can't understand how $q_\lambda(\psi_n-\psi_m)$ behaves.
Any help would be greatly apppreciated!
This is a bounded form: $0 \le q_{\lambda}(f) \le \|f\|^{2}$, regardless of $\lambda$, and it is defined everywhere on the Hilbert space. So the form norm $q_{\lambda}(f)+\|f\|^{2}$ and the usual norm are equivalent norms.