In Quantum Mechanics, my understanding is that we have a Hilbert space.
If we to model a particle in space we consider the space defined by the basis
$$|x\rangle$$
for each $x \in \mathbb{R}$
We then define the position operator to be the operator such that
$$\hat{x} |x\rangle = x|x\rangle$$
We can then construct a Hamiltonian operator, for example the SHO Hamiltonian
$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2$$
This leads us to energy eigenbasis $|n\rangle$ for $n \in \mathbb{N}$
But this has a different cardinality to the position eigenbasis (one is countable the other is not). So how can they both be basis for the same Hilbert space?
