quantum mechanics - Wave function and Dirac bra-ket notation

Would anyone be able to explain the difference, technically, between wave function notation for quantum systems e.g. $\psi=\psi(x)$ and Dirac bra-ket vector notation?

How do you get from one to the other formally?

When you express a state in bra ket notation as a vector $(a,b)$ for instance do we have to be referring to a basis of eigenvectors for an observable?

Furthermore, consider if we had a unitary function $U$ how would we express $\langle U \psi\mid$ in terms of $\langle \psi\mid$? As the bra of psi is an element of the duel Hilbert space, which is a function that takes the ket of psi to the inner product, how would we remove the unitary operator?

Note, this is not a homework question, i'm just trying to improve my formal understanding of the notation I've been using, as this has been skirted over quite substantially.

EDIT: Got the last question I asked $$\langle U\psi \mid \phi \rangle=\langle \psi \mid \bar{U^T}\phi \rangle$$ $$\forall \phi \in H \Rightarrow \langle U\psi\mid = \langle \psi\mid\bar{U^T}.$$

Answer

Here's how you get from one to the other. Let me take the case of a particle moving in one dimension.

In this case, we assume that there exist vectors $|x\rangle$ which form a "dirac-normalized" basis (the position basis) for the Hilbert space in the sense that their inner products satisfy $$ \langle x|x'\rangle =\delta(x-x') $$ Note, as an aside, that these vectors are not normalizable in the standard sense, and therefore they do not strictly speaking belong to the Hilbert space.

Next, for each $|\psi\rangle$ in the Hilbert space, we define the position basis wavefunction $\psi$ corresponding to the state $|\psi\rangle$ as $$ \psi(x) = \langle x|\psi\rangle $$ So really, the value $\psi(x)$ of the position basis wavefunction $\psi$ at a point $x$ can simply be thought of as the basis component of $|\psi\rangle$ in the direction of $|x\rangle$ just as in the finite-dimensional case where one can find the component of a vector $|\psi\rangle$ along a basis vector $|e_i\rangle$ simply by taking the inner product $\langle e_i|\psi\rangle$.