Let $\lambda^2$ be the set {$\phi:\phi$ is progressive and E[$\int_0^\infty \phi_s^2ds]< \infty$).
Let $\xi$ the space of simple process.
I know that the space $\xi$ is dense in ($\lambda^2, || . ||_{\lambda^2} )$.
How could I prove Ito's isometry also for the class of $\lambda^2$ integrands? (I already saw the proof for simple process).
I watched in several book but often they just say that is a direct consequence, without actually proving it.