I'm currently following a course in quantum mechanics that uses Griffith's textbook.
Griffiths shows that wave functions are members of a Hilbert space. Since this is an abstract vector space, we can assign a state vector $|{\psi}>$ to each wave function $\psi(x)$. Since our course on linear algebra runs pretty deep, this I understand.
Griffiths subsequently defines the inner product between two state vectors $<\phi|\psi> = \int\phi^*(x)\psi(x)dx$, and he derives all sorts of fun things from this inner product; that the function $\psi(x)$ corresponding to $|\psi>$ is the wave function as seen previously in wave mechanics, and that its Fourier transform determines the probability density of momentum.
At this point, something happens in both Griffiths and my course that I don't quite comprehend. Both my instructor and the book mention that at this point, the vectors will take precedence above the functions; all quantum mechanical information is from now on fully and completely recorded into the vector. The functions are simply a representation of this vector; one of the many possible representations. Am I correct in saying this?
But Griffith's proof that the coefficients in the position basis of $|\psi>$ form the values of the function $\psi(x)$ is completely based on the definition $<\phi|\psi> = \int\phi^*(x)\psi(x)dx$. How should I then read this? Does he assume that there is a function representation $\psi(x)$ corresponding to the state vector $|\psi>$ (which is in his right, since $L_2$ is isomorphic to the vector space), to subsequently derive the properties of this function?
To me it just seems really strange that we have these abstract state vectors, only to define the inner product in terms of one very specific representation of that vector.
Edit 6 February 2019: I can (for obvious reasons) not unmark this question as a duplicate, but I can "edit to explain why my question hasn't been answered before." My question is not about whether the ket psi and the wave function are the exact same--I have written in my question that I understand that the wave function is a representation of the ket (in other words, I've used the answer to the duplicate question to ask my own question). My question is why in so many derivations, it appears that the wave function formulation takes precedence (in e.g. the definition of the inner product). In the answers below, it has been explained that there is no precedence, and that this is simply a shortcut in the theory.
\rangle
to get more appropriate-looking ket's:$\vert\psi\rangle$
gives $\vert\psi\rangle$. $\endgroup$