let $\mathcal{H}^2$ be the space of uniform integrable martingales M s.t. $sup E[M_t^2]<\infty$. Define $(M,N)_{\mathcal{H}^2}=E[M_\infty N_\infty]$. Then $\mathcal{H}^2$ is a Hilbert space with this scalar product.
I don't get this statement. The author starts with that there is a bijection between the elements $M\in\mathcal{H}^2$ and their terminal values $M_\infty$. I even don't get this.