In QM a system is represented by a Hilbert Space rather than a classical Phase Space. So, system A might be described by Hilbert Space Ha and system B might be described by Hilbert Space Hb. Mathematically, Hilbert Spaces are many things, but the first thing they are, at the most fundamental level, is a set (of continuous, complex-valued, square-integrable functions, etc.). My Question: In QM what defines the scope of the Hilbert Space for a specific system? (What elements are in its Hilbert Space set?). As always, thanks in advance.