So one of the postulate of quantum mechanics is that observables have complete eigenfunctions. Can someone let me know if I am understanding this properly: Basically you postulate for example, position kets |x> such that any state can be represented by a linear combination of these states (integral), and you postulate an operator x' such that x'|x>=x|x>. So basically the Hilbert space is the span of |x>?...Then you can postulate other kets, like momentum kets |p> that also span the Hilbert space, and you postulate an operator p' such that p'|p>=p|p>. Then any state can be represented as an integral over |p>, including the position states. So by postulation, the span of |x> equals the span of |p>? Then we postulate the Schrodinger equation, and write H' as some combination of the x' and p' operators, and that the eigenfunctions of H' (which can be written in terms of |x> or |p>) span the same space spanned by |x> and |p>?...How do we know this? Or do instead the eigenfunctions of H' define the Hilbert space, and there are states that can be represented by linear combinations of |x> or |p> but aren't actual states?