Hey, I have been churning the idea of Dirac notation around in my head and I am thinking about the position and momentum basis representation of a wavefunction in a Hilbert space. Wikipedia mentions the following in the article 'Bra-ket notation' under the heading 'Position-space wave function': <r|ψ> = ψ(r) by definition. Now, I can appreciate how there's can be many infinitely many wavefunctions for a quantum system and for each possible wavefunction, there is mapping from the position space onto the field of complex numbers. That's the idea encoded by the representation ψ(r) of the wavefunction, right? What I am having trouble trouble to understand is (r, ψ). ψ is, surely, an element of a Hilbert space that corresponds to the quantum system, but r is not an element from that Hilbert space. So, how can you have an inner product between the two? Any ideas?