I am given to believe that one way that one would could represent a wavefunction is by the expansion
$$\Psi(x) = \Sigma_n \Psi_n(x) = \Sigma_n f_n\phi_n(x) \tag{1}$$
where $\{\phi_n (x) \}$ is an orthonormal base of functions.
$$f_n = proj_{\phi_n} \Psi_n(x) \tag{2}$$
Firstly, is this correct? Or is the projection that is described supposed to be the following:
$$ f_n = proj_{\phi_n} \Psi(x) \tag{3} $$
Secondly, I am also given to believe that one way one can represent the projection described is
$$ f_n = \langle \phi_n(x),\Psi_n(x) \rangle \tag{4} $$
in Dirac notation. So, then when I am projecting functions onto basis functions and the basis functions are not normalized while the wavefunctions are, what is the appropriate expression to describe the projection? Is it like the equation for vectors, namely:
$$ proj_u v = \frac{u \cdot v}{u \cdot u} u \tag{5}$$
Like how is it expanded to functions?