I'm trying to work through Schumacher and Westmoreland's Quantum Processes, Systems, and Information, and in their discussion on conditional states (6.4) they introduce the partial inner product which has me quite stumped.
Consider a composite quantum system AB consisting of two subsystems A and B. The subsystems are described by Hilbert Spaces $\mathcal{H}^{(\mathrm{A})}$ and $\mathcal{H}^{(\mathrm{B})}$ respectively, and AB is described by $\mathcal{H}^{(\mathrm{AB})} = \mathcal{H}^{(\mathrm{A})} \otimes \mathcal{H}^{(\mathrm{B})}$. Let $|\alpha^{(\mathrm{A})}\rangle \in \mathcal{H}^{(\mathrm{A})}$ and $|\Phi^{(\mathrm{AB})}\rangle \in \mathcal{H}^{(\mathrm{AB})}$ be arbitrary vectors. Then $|\phi_\alpha^{(\mathrm{B})}\rangle = \langle \alpha^{(\mathrm{A})} | \Phi^{(\mathrm{AB})}\rangle$ (the partial inner product) is defined to be a vector in $\mathcal{H}^{(\mathrm{B})}$ such that, for all $|\beta^{(\mathrm{B})}\rangle \in \mathcal{H}^{(\mathrm{B})}$,
$$ \langle \beta^{(\mathrm{B})} | \phi_\alpha^{(\mathrm{B})}\rangle = \langle \alpha^{(\mathrm{A})}, \beta^{(\mathrm{B})} | \Phi^{(\mathrm{AB})}\rangle.$$
I am first asked to show the existence and uniqueness of $|\phi_\alpha^{(\mathrm{B})}\rangle$. I think I understand how to do this, but I would appreciate any comments on it nonetheless.
Then I am asked to show that, given $|\Psi^{(\mathrm{AB})}\rangle \in \mathcal{H}^{(\mathrm{AB})}$ and an orthonormal A-basis $\{ |a^{(\mathrm{A})}\rangle \}$,
$$ |\Psi^{(\mathrm{AB})}\rangle = \sum_a |a^{(\mathrm{A})}\rangle \otimes \langle a^{(\mathrm{A})} |\Psi^{(\mathrm{AB})}\rangle, $$
and have no idea how to even begin.
In general, if you have any advice for how to think about the partial inner products (and conditional states as well), it would be greatly appreciated. Thanks!