In the literature I have seen that the notions "our system is continuous variable system", "Hilbert space of our system is infinite" were used as if they were equivalent.
For example for harmonic oscillator the Hilbert space is infinite, but I can write my system in discrete basis (with number operator eigenvectors) or continuous basis (position operator eigenvectors).
So I'm a little confused:
What is the definition of "continuous variable system"?
Is harmonic oscillator continuous variable system?